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Эта публикация цитируется в 6 научных статьях (всего в 6 статьях) Attractors. Then and now S. V. Zelikabc a Zhejiang Normal University, Department of Mathematics, Zhejiang, P. R. China b University of Surrey, Department of Mathematics, Guildford, United Kingdom c Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow, Russia
Англоязычная версия:
Russian Mathematical Surveys, 2023, Volume 78, Issue 4, Pages DOI: https://doi.org/10.4213/rm10095e 
Реферативные базы данных:
    1. IntroductionOne of the most surprising lessons of dynamical systems theory is that even relatively simple and deterministic ordinary differential equations (ODEs) may demonstrate a very complicated behaviour with strong random features (a so-called deterministic chaos). This phenomenon has intensively been studied starting from the second part of the 20th century, a lot of prominent results have been obtained in this direction, and a lot of methods for investigating it have been developed (such as hyperbolic theory, Lyapunov exponents, homoclinic bifurcation theory, strange attractors, and so on; see [62], [91], [116], [202], [210], and the references therein). However, this phenomenon occurred to be much more complicated than it was thought from the very beginning, so, despite many efforts made, the existing theory is mainly limited to low-dimensional model examples (even in these examples, it is established that, in some cases, the problem of a full description of the dynamics is not algorithmically solvable) and we still do not have really effective methods for studying the dynamical chaos in higher-dimensional systems of ODEs. The situation becomes even more complicated when we deal with dynamical systems generated by partial differential equations (PDEs). For such systems, the initial phase space is infinite-dimensional (for example, $L^2(\Omega)$, where $\Omega$ is a domain in $\mathbb{R}^d$), which causes a lot of extra difficulties. In addition, together with a temporal variable $t$ and related temporal chaos, we now have spatial variables $x\in\Omega$, so the spatial chaos may naturally appear. Also, we may have an interaction between the spatially and temporal chaotic modes which form the so-called spatio-temporal chaos. As a result, a new type of purely infinite-dimensional dynamics with a principally new level of complexity (unreachable in systems of ODEs studied in classical dynamics) may arise; see [174], [221], [239], and the references therein. Nevertheless, there exists a wide class of PDEs, namely, the class of dissipative PDEs in bounded domains, where, despite the infinite-dimensionality of the initial phase space, the effective limit dynamics is finite-dimensional in a certain sense. In particular, this dynamics can be described using finitely many parameters (the “order parameters” in the terminology of I. Prigogine; see [187]), whose evolution obeys a system of ODEs (which is called an inertial form of the initial dissipative system). As hinted by their name, dissipative systems consume energy (in contrast to conservative systems where the total energy is usually preserved), so, in order to have non-trivial dynamics, the energy income should be taken into account (in other words, the system under consideration should be open and should interact with the external world). On the physical level of rigour, the rich and complicated dynamical structures (often referred to as dissipative structures following Prigogine) arise as the result of interaction of the following three mechanisms: 1) energy decay, usually more essential in higher “Fourier modes”; 2) energy income, usually through the lower “Fourier modes”; 3) energy flow from lower to higher “modes” which is provided by the nonlinearities. Moreover, it is typical for dissipative PDEs in bounded domains that the number of lower “modes” where the energy income is possible is finite. So, it is natural to expect that these modes are, in a certain sense, dominating and the higher modes are slaved by the lower ones. This supports the conjecture that the effective dynamics in such systems is finite-dimensional up to some transient behaviour (where the unstable lower “Fourier modes” are treated as order parameters) and somehow explains why the ideas and techniques of classical finite-dimensional theory of dynamical systems are also effective for describing the dissipative dynamics in such PDEs. However, the above arguments are very non-rigorous from the mathematical point of view, and it is extremely difficult/impossible to make them meaningful. It is even unclear what are the “modes” in the above statements. Indeed, they are rigorously defined as Fourier/spectral modes in the linear theory and a priori make sense only for systems somehow close to linear ones. Moreover, such modes are natural for the linearization near equilibria only and may not exist at all when the linear equations with time-periodic coefficients are considered. Thus, despite the common usage of modes and related length scales for highly nonlinear systems in physics literature (for example, in turbulence, see [81] and the references therein), the precise meaning of them is usually unclear. That is the reason why the mathematical theory of dissipative PDEs is based on related but different concepts which at least can be defined rigorously. Namely, let a dissipative system be given by a semigroup $S(t)$, $t\geqslant 0$, acting in a Hilbert, Banach (or sometimes even Hausdorff topological) space $\Phi$. Usually, in applications $\Phi$ is some infinite-dimensional function space where the initial data live (say, $\Phi=L^2(\Omega)$) and $S(t)$ is a solution operator of the PDE under consideration. Then the dissipativity is often understood as the validity of the following dissipative estimate:
$$
\begin{equation}
\|S(t)u_0\|_\Phi\leqslant Q(\|u_0\|_\Phi)e^{-\alpha t}+C_*
\end{equation}
\tag{1.1}
$$
for appropriate positive constants $C_*$ and $\alpha$ and monotone increasing function $Q$ which are independent of $u_0$ and $t$. Roughly speaking, this estimate shows that there is a balance between energy injection and dissipation, so the energy cannot grow to infinity. Moreover, for very high energy levels dissipation is dominating and the energy decays. As we have already mentioned, the class of dissipative PDEs is rather wide and includes many physically relevant examples, such as the Navier–Stokes equations, reaction-diffusion equations, damped wave equations, various pattern formation equations, and so on (see, for example, [215] and the references therein for more examples). The central concept of the theory is a (global) attractor $\mathcal A$. By definition, it is a compact subset of the phase space $\Phi$, which is invariant with respect to the semigroup $S(t)$ and attracts the images (under the map $S(t)$) of all bounded sets as time tends to infinity. Thus, on the one hand the attractor $\mathcal A$ consists of the full trajectories of the dynamical system under consideration and contains all of its non-trivial dynamics. On the other hand, it is essentially smaller than the initial phase space. In particular, the compactness assumption shows that the higher modes are indeed suppressed, so even the existence of a global attractor supports somehow the conjecture of finite-dimensionality. The ultimate goal of the present survey is to develop a unified approach to the theory of attractors for dissipative dynamical systems generated by PDEs and to clarify its role in the rigorous justification of the above-mentioned conjecture of finite-dimensionality. We emphasize from the very beginning that, although the study of such systems is also strongly based on the analysis of PDEs (for instance, the well-posedness of the 3D Navier–Stokes problem is one of the most challenging open problems in the theory of PDEs, and the absence of a reasonable answer whether and in what sense this system is well posed clearly affects the attractor theory for the Navier–Stokes equations), a more or less detailed exposition of the modern methods of the analysis of PDEs is far beyond the scope of this survey, where we concentrate mainly on various dynamical concepts of the theory of attractors. The systematic study of the dynamical properties of dissipative PDEs started, to the best of our knowledge, from the pioneering papers of Foias and Prody [75] and Ladyzhenskaya [147] and has been motivated by the dream to understand the turbulence using the methods of the theory of dynamical systems for ODEs. In particular, the existence of an invariant measure for the Navier–Stokes system, as well as finitely many determining modes, was established in [75] and the object which coincides with the modern attractor for the Navier–Stokes problem was implicitly introduced in [147]. We mention also [16], where the global attractors were introduced with applications to delay differential equations. The theory of attractors has been intensively developing during the last 50 years and many interesting and promising results have been obtained; see [12], [8], [38], [43], [44], [93], [149], [174], [192], [215], [223], [243], and the references therein. In particular, one of the main results of the attractor theory claims that, under the mild assumptions on the dissipative system generated by an evolutionary PDE in a bounded domain, the corresponding attractor $\mathcal A$ has a finite fractal dimension: Combining this fact with Mané’s projection theorem, we get a one-to-one homeomorphic projection of the attractor $\mathcal A$ onto a finite-dimensional set $\overline{\mathcal A}\subset\mathbb{R}^N$. Moreover, the reduced dynamics on $\overline {\mathcal A}$ is governed by a system of ODEs. Thus, the existence of a finite-dimensional attractor allows us to build up an inertial form of the dissipative PDE under consideration and justify somehow the finite-dimensionality conjecture; see [193], [243], and the references therein for more details.However, this justification is not entirely satisfactory since the inertial form obtained is only Hölder continuous, no matter how smooth the initial PDE is, so, starting from a smooth dynamical system, we end up with non-smooth equations where even the uniqueness of a solution may be lost. Any attempts to improve the regularity of this reduction lead to extra restrictive assumptions on the initial PDE which are hard to check and which are violated in many interesting examples. Moreover, as recent examples show, this problem is far from being technical since, despite the finiteness of the fractal dimension of the global attractor and the existence of a Hölder continuous inertial form, the associated limit dynamics may remain, in a sense, infinite-dimensional and demonstrate features which are not observable in classical dynamics generated by smooth ODEs (like limit cycles with super-exponential rate of attraction, decaying travelling waves in Fourier space, and so on); see [64], [133], [243], and the references therein. These examples allow us to guess that the sharp borderline between finite and infinite-dimensional limiting dynamics is more related to Lipschitz-continuous inertial forms and inertial manifolds rather than to Hölder-continuous inertial forms and the fractal dimension of the attractor. Thus, despite many efforts made, the rigorous interpretation and justification of the finite-dimensional reduction in dissipative systems remains a mystery and is one of the most challenging problems in the modern theory of attractors. We discuss below in more details some of the currently known approaches to tackle this and related problems. The survey is organized as follows. In Section 2 we give a flavour of attractor theory by considering the simplest low-dimensional examples, where the attractor can be found more or less explicitly, and we demonstrate the difference between various types of attractors, their dependence on parameters, their dimensions, and so on. We hope that this section will help the reader with understanding the more general and abstract theory presented in the next sections. Section 3 is one of the central sections of the survey, where we develop the attractor theory in general Hausdorff topological spaces, which, in turn, allows us to build up a unified approach to different types of attractors. Namely, to define an attractor, we need to specify first what objects will be attracted by this attractor. We refer to the system of these objects as a bornology $\mathbb B$ on the phase space of the problem, keeping in mind that in the standard theory a (global) attractor usually attracts bounded subsets of the phase space. Next, we should specify in what sense (in what topology) the attraction will hold. In other words, we need to specify the topology on the phase space $\Phi$. Surprisingly, a substantial attractor theory, which is similar to the standard one, can be constructed in the general situation where $\Phi$ is just a Hausdorff topological space, under some minimal assumptions on the bornology $\mathbb B$. Although attractors in general topological spaces were studied before (see [38], [86], [170], and the references therein), many results in Section 3 are hard/impossible to find in the literature in the necessary generality, so we present more or less detailed proofs for most of them. In Section 4 we apply the unified approach presented above to the case where the PDE under consideration does not possess the unique solvability property or this uniqueness of a solution is not known yet. One of the most natural approaches to tackle this type of problems (proposed by Chepyzhov and Vishik in [36], see also [38], and by Sell [208]) is related to constructing the trajectory dynamical system associated with the PDE under consideration and studying its attractors. Roughly speaking, we replace the initial phase space $\Phi$, where the problem is ill posed and the corresponding solution semigroup can be defined as a multi-valued semigroup only, by the new phase space $\mathcal K_+$ which consists of all positive semi-trajectories of our PDE satisfying some nice properties. Then, if $\mathcal K_+$ is chosen in the proper way, the semigroup $T_h$, $h\in\mathbb{R}_+$, of temporal shifts (defined by $(T_hu)(t):=u(t+h)$) acts on $\mathcal K_+$ and $(T_h,\mathcal K_+)$ is exactly the trajectory dynamical system associated with the problem under consideration. Note that, in the case where the uniqueness property holds, this semigroup is conjugate to the usual solution semigroup $S(t)$ acting in the initial phase space $\Phi$, so the theory is consistent. The advantage of this approach is that the construction of the trajectory dynamical system does not require the uniqueness property to hold, so we may study its attractors in a usual way, avoiding the use of multi-valued maps. The only problem is to define a bornology and topology on $\Phi$ in the proper way. In relatively simple situations, both topology and bornology can be lifted from the initial phase space, but in more complicated cases (like the 3D Navier–Stokes equations) this does not work. Moreover, there are several alternative inequivalent ways to do this, so the unified approach introduced in Section 3 works in full strength here. The purpose of Section 4 is, in particular, to give the comparison of known alternative approaches to the trajectory attractors using the example of the 3D Navier–Stokes equations. To the best of our knowledge, this has never been done before. Section 5 extends the unified approach to attractors in the non-autonomous case. We consider the most general case, where phase space for the dynamical process $U(t,\tau)$, $t\geqslant \tau$, associated with the problem under consideration may depend on time, so that $U(t,\tau)\colon\Phi_\tau\to\Phi_t$ and $\{\Phi_t\}_{t\in\mathbb{R}}$ is a family of Hausdorff topological spaces, and we use the pullback attraction property to define attractors. In this case an attractor is understood as a time-dependent set $\mathcal A(t)\subset\Phi_t$, $t\in\mathbb{R}$, the bornology $\mathbb B$ (which is often referred to as a universe in this theory) also consists of time dependent sets $B(t)\subset\Phi_t$, $t\in\mathbb{R}$, and the attraction property is pullback in time, that is, if you fix $t\in\mathbb{R}$ and start from $B(\tau)\in\mathbb B$, $\tau<t$, then the image $U(t,\tau)B(\tau)$ is close to $\mathcal A(t)$ if $t-\tau$ is large enough (see Section 5 for the details). The particular case where $\Phi_t$ are Banach spaces and $\mathbb B$ consists of uniformly (in time) bounded sets was considered in [50]; the extension proposed allows us to treat from the unified point of view also the case of the bornology of tempered sets (which is important for random attractors), as well as many other interesting examples. This general theory covers, in particular, the case of cocycles (or skew-products). We recall that a family of maps $\mathcal S_\xi(t)\colon\Phi\to\Phi$, $\xi\in\Psi$, $t\geqslant 0$, is a cocycle over a group $T(h)\colon\Psi\to\Psi$, $h\in\mathbb{R}$, acting on a topological space $\Psi$ if
$$
\begin{equation*}
\mathcal S_\xi(0)=\operatorname{Id},\qquad \mathcal S_\xi(t+h)=\mathcal S_{T(h)\xi}(t)\circ\mathcal S_\xi(h),\quad t,h\geqslant 0.
\end{equation*}
\notag
$$
These objects are natural for the theory of non-autonomous and random dynamical systems. Roughly speaking, the underlying group $T(h)\colon\Psi\to\Psi$ describes the evolution of the time-dependent symbol of the non-autonomous PDE in question under time shifts and the $\mathcal S_\xi(t)$ are the solution operators (from zero moment of time to time $t$) of the PDE with the given symbol $\xi\in\Psi$; see [27], [35], [38], [96], [120], [121], and the references therein. Recall that the dynamical process which corresponds to the symbol $\xi$ can be recovered from the cocycle using the simple formula $U_\xi(t,\tau)=\mathcal S_{T(\tau)\xi}(t-\tau)$, and exactly this relation allows us to extend the theory from dynamical processes to cocycles. Note also that introducing a Borel probability measure on $\Psi$, which is invariant (and usually ergodic) with respect to the group $T(h)$, allows us to link this theory with the theory of random dynamical systems and their attractors; see [3], [27], [52], [54], [58], [121], and the references therein. In Section 5 we also discuss an alternative approach to attractors of non-autonomous dynamical systems, which is based on the reduction of the cocycle related to the PDE under consideration to the autonomous semigroup acting on the extended phase space $\Phi\times\Psi$. This approach leads to an object which is independent of time and the rate of attraction to it is uniform in time as well as with respect to $\xi\in\Psi$; for this reason it is referred to as a uniform attractor. We present here the classical results related to weak uniform attractors and to strong ones (in the case of translation-compact external forces): see [37], [38] and the references therein, as well as more recent results concerning strong uniform attractors in the case of non-translation-compact external forces: see [244] and the references therein. Section 6, which is the second central section of the survey, discusses the dimensions of attractors. As we have already mentioned, the fact that the attractor $\mathcal A$ of the PDE under consideration has a finite fractal dimension allows us to build up an inertial form for this equation, and this is one of the possible ways to justify the finite-dimensionality conjecture for the associated limit dynamics, so getting realistic upper and lower bounds for this dimension is one of the most fashionable branches of the attractor theory. This activity was originated by the seminal paper of Mallet-Paret [165] (see also [148]), where a method for estimating the dimension of negatively invariant sets, which is based on some smoothing/squeezing properties for differences of solutions, was proposed. In a modern interpretation, the main result can be formulated as follows: let $\Phi$ and $\Phi_1$ be two Banach spaces such that $\Phi_1$ is compactly embedded in $\Phi$, and let $B$ be a bounded set in $\Phi_1$ which is negatively invariant with respect to some map $S$. Assume that $S$ enjoys the following kind of squeezing property:
$$
\begin{equation}
\|S(u_1)-S(u_2)\|_{\Phi_1}\leqslant\kappa\|u_1-u_2\|_{\Phi_1}+ L\|u_1-u_2\|_\Phi\qquad \forall\,u_1,u_2\in B
\end{equation}
\tag{1.2}
$$
for some $\kappa\in[0,1)$ and $L>0$. Then the fractal dimension of $\mathcal A$ in $\Phi_1$ is finite (see, for example, [71] or [46]). This result and various generalizations of it to non-autonomous and random cases are presented in Section 6. Note that this approach has a tremendous number of applications in the modern theory of attractors (see, for example, [46], [65], [129], [163], [164], [238], and the references therein); in particular, most results concerning exponential attractors are strongly based on this method; see the discussion below. We also discuss there the volume contraction method for obtaining upper bounds for the Hausdorff and fractal dimension of an attractor which was proposed by Douady and Oesterlé [60] (see also [107] and [49] for generalizations to the infinite-dimensional case). Roughly speaking, the key result here is that if you have a negatively invariant (with respect to some smooth map $S$) compact set $B$ of a Hilbert space, such that the infinitesimal $k$-dimensional volumes in it are contracted by $S$, then the dimension of $B$ is less than $k$. In the case of the Hausdorff dimension this result was established in [60] and [107] for the finite and infinite-dimensional cases, respectively. The case of fractal dimension is more delicate and for a long time only partial results with extra unnecessary assumptions were known (see [38], [49], and the references therein). The breakthrough there was done by Hunt [105], where exactly the same result was established for the fractal dimension in the finite-dimensional case. This result was subsequently extended to the infinite-dimensional case in [19] under some extra technical assumptions, which were finally removed in [32]. The advantage of this method is that, combined with the use of the Lieb–Thirring inequalities presented by Lieb in [155], it gives the best known upper bounds for the fractal dimension of the attractor of the 2D Navier–Stokes equations: see, for example, [215], see also [79] and [32] for the best analytic bounds for this dimension via the Lieb–Thirring inequalities. This makes the volume contraction method very popular in attractor theory (despite the fact that it also has drawbacks and is not applicable in many cases, in particular, when the solution operators are not differentiable with respect to the initial data; see [179] and the references therein). We give an exposition of this method in Section 6 too. We also discuss lower bounds for the dimensions of attractors in that section. Recall that the most widespread method to get such estimates is based on the fact that an unstable manifold of any (hyperbolic) equilibrium always belongs to the attractor, so its dimension cannot be smaller than the dimension of this unstable manifold. This gives us lower bounds for the dimension of the attractor in terms of the instability indices of equilibria (see [12], [215], and the references therein). However, this is not enough for sharp lower bounds in some cases and there are examples where the instability indices of all equilibria remain bounded, but the dimension of the attractor tends to infinity as the physical parameter of the system under consideration tends to zero. For this reason we discuss also a more exotic but promising alternative method (proposed by Turaev and Zelik [220]), which is based on homoclinic bifurcation theory. Roughly speaking, this method relies on the fact that, under some natural assumptions, an invariant torus of very high dimension (which is restricted by the Lyapunov dimension of the corresponding equilibrium) may bifurcate from a homoclinic orbit under an appropriate choice of the perturbation. This allows us to connect lower bounds with the Lyapunov dimension (similarly to the upper bounds obtained via the volume contraction method). This is very important for weakly dissipative equations and gives us sharp upper and lower bounds of the same order for some class of damped wave equations. In Section 7, we discuss the theory of inertial manifolds for dissipative PDEs. Recall that, by definition, an inertial manifold is a smooth invariant finite-dimensional submanifold of the phase space of the problem under consideration which is globally stable and normally hyperbolic. The existence of such a manifold provides a perfect justification for the conjecture of finite dimensionality of limit dynamics. Indeed, the restriction of our system to the inertial manifold gives the desired smooth inertial form which is governed by a system of ODEs on the base of the manifold. On the other hand, due to normal hyperbolicity we have the exponential tracking (asymptotic phase) property, which guarantees that any other trajectory of the PDE under consideration attracts exponentially to the corresponding trajectory on the inertial manifold, so we do not lose any important information about the limit dynamics after going over to the reduced inertial form on the manifold. To the best of our knowledge, such an object was first constructed by Mané in [168] in the case of a reaction-diffusion equation and became popular after [76], where this result was extended to more general class of equations and was associated with the dream to understand turbulence (even the attribute “inertial” introduced there was motivated by the inertial scale in the conventional theory of turbulence; see, for example, [81] and the references therein). The classical construction of an inertial manifold requires the PDE under consideration to satisfy rather restrictive conditions, which allow one to present the associated dynamical system as a slow-fast system and slave the fast modes to slow ones using the standard methods of hyperbolic theory (see [76], [124], [175], [193], [195], [199], [209], [215], [243], and the references therein). These conditions are usually formulated in terms of spectral gap conditions on the leading linear part of the equation under consideration. On the one hand, these conditions are not satisfied, for instance, in the case of the 2D Navier–Stokes equations, so the existence or non-existence of an inertial manifold for the Navier–Stokes equations is one of the most challenging open problems of the theory. On the other hand, it is also known that the spectral gap conditions are sharp, at least on the level of abstract functional models (for example, in the class of abstract semilinear parabolic equations; see [64], [243]) so a further progress in constructing inertial manifolds beyond the spectral gap conditions is possible only by exploiting some special properties of concrete classes of PDEs. The first result in this direction was obtained by Mallet-Paret and Sell in [166], where inertial manifolds were constructed in the case of scalar reaction-diffusion equations in 3D with periodic boundary conditions, by means of the method of spatial averaging; see also [167] where it was shown that this method does not work in spaces of dimension higher than three. Taking into account the recent progress in this area (for instance, the extension of methods of spatial averaging to the 3D Cahn–Hilliard equation: see [131], various truncated or regularized versions of the 3D Navier–Stokes equations, see [82], [126], [128], the development of the method of spatio-temporal averaging and its applications to the 3D complex Ginzburg–Landau equation: see [127], [130]), we give a brief exposition of this method in Section 7. An alternative method for constructing inertial manifolds is based on transforming the initial PDE or embedding it into a new system of PDEs in such a way that the obtained new system satisfies the spectral gap conditions. This method was originally related to an erroneous attempt of Kwak [144] to prove the existence of an inertial manifold for the 2D Navier–Stokes equation; see also [145], [216], and see [134] for a clarification of the nature of that error. For this reason, the whole method was forgotten for a long time and was considered as suspicious and potentially erroneous. The situation has changed recently after the works [126] and [132], where an appropriate modification of this method was been used to solve the long-standing open problem about the existence of inertial manifolds for general 1D reaction-diffusion-advection systems (see also [230]–[232] for some preliminary results in this direction). We include a brief exposition of this alternative method in Section 7. We also discuss the smoothness of inertial manifolds. It is known that, in general, even when the initial system is analytic, the inertial manifold related to this PDE is only $C^{1+\varepsilon}$-smooth for some small $\varepsilon>0$ and a further regularity of inertial manifolds requires much stronger versions of spectral gap conditions, which are not satisfied even in the simplest examples (see [40], [124]). This looks like a big drawback of the theory since the regularity of an inertial manifold is important from both the theoretical and applied points of view. Indeed, even the analysis of simplest bifurcations requires a higher smoothness than $C^{1+\varepsilon}$ (for example, for an analysis of the Andronov–Hopf bifurcation, we need $C^3$; see, for instance, [210]). On the other hand, the low regularity of the inertial manifold and the related inertial form prevents us from using higher-order numerical methods. This problem looked unsolvable for a long time, but as shown very recently (see [136]) it nevertheless can be overcome. Namely, by increasing the dimension of the manifold and using a clever cut-off procedure (based on Whitney’s extension theorem) we may kill resonances and other obstacles to the existence of smooth invariant manifolds and get $C^k$-smooth inertial manifolds for every finite $k$. Section 8 of the survey is devoted to exponential attractors. These objects were introduced in [63] as, in a certain sense, intermediate objects between usual attractors and inertial manifolds, in order to overcome key drawbacks of attractor theory. The main of these drawbacks is exactly the slow rate of attraction and, which is even more important, the fact that it is impossible, in a more or less general situation, to control this rate of attraction in terms of physical parameters of the PDE under consideration. This makes the attractor, in a sense, unobservable in experiments: no matter how long we wait, we can never be sure that we are close to the attractor. The absence of such a control also leads to the sensitivity of the attractor to perturbations. Roughly speaking, the idea of the construction of an exponential attractor, is to add some special points (for example, metastable states) to the usual attractor in such a way that, on the one hand, the rate of attraction to the new object becomes exponential and controllable and, on the other hand, the size of this object does not grow too much: in particular, it should still have a finite fractal dimension, so that Mané’s projection theorem still allows us to construct an inertial form on it. Following [65], the modern theory of exponential attractors is based on the squeezing property (1.2) and its various generalizations (see [9], [44], [65], [68], [69], [72], [84], [179], and the references therein). Thus, in contrast to inertial manifolds, exponential attractors are as general as the usual finite-dimensional global attractors (see also [67], [66] for infinite-dimensional exponential attractors in the case where the dimension of a usual attractor is infinite). We do not present the most general conditions of the form (1.2) which guarantee the existence of exponential attractors (we refer the interested reader to the surveys [67] and [179]). Instead, we discuss the impact of the theory of exponential attractors on non-autonomous and random attractors. Our exposition follows mainly [71] and [211] and is based on the straightforward extensions of (1.2) to the non-autonomous case. We recall that the theory of exponential attractors allows us to overcome the extra drawbacks of the theory of usual attractors which arise when the non-autonomous case is considered. Namely, again because of the non-controllable and non-uniform rate of attraction to, say, pullback attractors, we lose in general attraction forward in time. The situation is a bit better in the case of random attractors, where we have attraction in probability forward in time, but the lack of the uniformity causes a lot of difficulties, in particular, it does not allow us to establish the robustness of random attractors in the random-deterministic limit (see, for example, [42], [54] and also [24] and the references therein for the contemporary approach to this problem). In contrast to this, non-autonomous exponential attractors can be constructed in such a way that the rate of attraction is uniform and exponential both forward and pullback in time and this remains true for random exponential attractors as well (see Section 8 for the details). Since the sensitivity of attractors with respect to perturbations is closely related to the rate of attraction, we also include some elements of perturbation theory of attractors to the survey, as well as a theory of so-called regular attractors in the terminology of Babin and Vishik (see [10]). Such attractors are typical for systems which possess a global Lyapunov function. Then, under the extra generic assumption that the number of equilibria $\mathcal R$ is finite and all of them are hyperbolic, the attractor consists of a finite union of finite-dimensional unstable manifolds of these equilibria, and every complete trajectory lying on the attractor is a heteroclinic orbit between these equilibria. These attractors possess a number of nice properties, and they have many similarities with exponential attractors. In particular, the rate of attraction to them is exponential and they are Hölder continuous with respect to perturbations (including non-autonomous ones); see [10], [12], [26], [94], [95], [228] and the references therein. In Section 9 we discuss an alternative approach to the problem of finite-dimensional reduction (which was proposed in [75], see also [147], and was historically the first) related to determining functionals. By definition, a system $\mathcal F:=\{\mathcal F_1,\dots,\mathcal F_N\}$ of continuous functionals $\mathcal F_i\colon\Phi\to\mathbb{R}$ on the phase space $\Phi$ is (asymptotically) determining for a semigroup $S(t)\colon\Phi\to\Phi$ if for any two trajectories $u_1(t)$ and $u_2(t)$ of this semigroup, the convergence
$$
\begin{equation*}
\lim_{t\to\infty}\bigl(\mathcal F_i(u_1(t))- \mathcal F_i(u_2(t))\bigr)=0,\qquad i=1,\dots,N,
\end{equation*}
\notag
$$
implies that $\lim_{t\to\infty}\|u_1(t)-u_2(t)\|_\Phi=0$. Thus, the asymptotic behaviour of trajectories of the system under consideration is determined by the behaviour of the finitely many quantities $\xi_i(t):=\mathcal F_i(u(t))$, $i=1,\dots,N$. Note, however, that determining functionals do not give a true finite-dimensional reduction since the quantities $\xi_i(t)$ do not obey a finite system of ODEs. Moreover, they typically satisfy some kind of a system of ODEs with delay (and realize a Lyapunov–Schmidt reduction), so that the phase space of the reduced system of ODEs with delay remains infinite-dimensional, and a true finite-dimensional reduction cannot actually be constructed in this way. Nevertheless, this topic remains interesting and fashionable (see [41], [46], [47], [77], [78], [184]) since determining functionals and the related reduction to delay ODEs have many important applications. For instance, they are used to establish the controllability of an initially infinite-dimensional system by finitely many modes (see, for example, [5]), to verify the uniqueness of an invariant measure for random/stochasitc PDEs (see, for example, [142]), and in data assimilation problems where the values of the functionals $\mathcal F_i(u(t))$ are interpreted as the results of observations and where the theory of determining functionals allows us to build up new methods for recovering the trajectory $u(t)$ from the results of observations; see [5], [4], [184], and the references therein. In our exposition we mainly follow the recent paper [114] and try to clarify the nature of determining functionals as well as the nature of the minimum number of such functionals (the so-called determining dimension $\dim_{\det}(S(t))$ of the system under consideration). The trivial lower bound for this dimension is related to the size of the set of equilibria $\mathcal R$, namely its embedding dimension $\dim_{\rm emb}(\mathcal R)$, that is, the smallest $N$ such that there is a continuous injective map $F\colon\mathcal R\to\mathbb{R}^N$. Surprisingly, this lower bound is sharp and we have the two-sided estimate
$$
\begin{equation*}
\dim_{\rm emb}(\mathcal R)\leqslant \dim_{\det}(S(t))\leqslant \dim_{\rm emb}(\mathcal R)+1
\end{equation*}
\notag
$$
(see [114] for the details). The proof of this result is based on the famous Takens delay embedding theorem and its generalizations for Hölder continuous maps (see [193], [203], [213], and the references therein). In particular, in the generic case where the set $\mathcal R$ is finite, almost every continuous function $\mathcal F\colon\Phi\to\mathbb{R}$ is a determining functional for the PDE under consideration, and such a determining functional can be chosen in the class of polynomial maps of sufficiently high degree. Thus, in contrast to a widespread paradigm, the minimum number of determining functionals describes the structure/size of the set of equilibria $\mathcal R$ of the system under consideration and is related neither to the complexity of the corresponding dynamics on the attractor, nor to its dimension, nor even to the dissipativity of the PDE under consideration. The corresponding examples, illustrating this statement, are also presented in Section 9. Finally, the function spaces most important for our purposes and their properties, which are used throughout the survey, are collected in the appendix (see Section 10). To conclude, we note that, because of the restricted size of this survey, it is not possible to pay the proper attention to all important works in the area of attractors, so the choice of material is somehow subjective and reflects the personal preferences of the author, and many interesting areas (like PDEs with delay, attractors in unbounded domains, approximate inertial manifolds, infinite-dimensional centre manifolds and the corresponding hyperbolic theory, and so on) are out of this survey. The author apologizes for the inconvenience caused. 2. Attractors: basic theory and model examplesThe aim of this section is to discuss briefly various concepts related to attractors and illustrate them by simple examples. We assume here that we are given a phase space $\Phi$ which will be a normed space for the moment (the more general situation where $\Phi$ is a Hausdorff topological space will be considered in what follows) and a semigroup $S(t)\colon\Phi\to\Phi$ satisfying
$$
\begin{equation}
S(t+h)=S(t)\circ S(h),\quad t,h\geqslant 0,\qquad S(0)=\operatorname{Id}.
\end{equation}
\tag{2.1}
$$
We refer to this semigroup $S(t)$ acting on $\Phi$ as a dynamical system and denote it by $(S(t),\Phi)$. Usually, the $S(t)$ are the solution operators of an ODE or an evolutionary PDE under consideration (which map the initial data to the corresponding solution at time $t$). Thus, we assume implicitly that the corresponding initial value problem is globally solvable and this solution is unique (although this concept may be used even in cases without uniqueness; see Section 4 for more details). The phase space $\Phi$ is often either some Sobolev space or its subspace endowed with an appropriate topology; see examples below. The key concept behind the attractor theory is the concept of an $\omega$-limit set. Definition 2.1. Let $B\subset\Phi$ be an arbitrary non-empty subset of $\Phi$. Then the $\omega$-limit set of $B$ is defined by where $[V]_\Phi$ stands for the closure of the set $V$ in the topology of $\Phi$. In the case of metric spaces, we may give an equivalent, but sometimes more convenient, sequential definition: In the general case where $\Phi$ is not metrizable, these two definitions may produce different objects. It is well known that, without further assumptions, an $\omega$-limit set can easily be empty or, even if it is occasionally not empty, it may not possess important properties like invariance or/and attraction. In order to preserve them, we need some kind of compactness, which is the central assumption of the attractor theory. We summarize the properties of an $\omega$-limit set in the following lemma. Lemma 2.2. Let the semigroup $S(t)$ be asymptotically compact on $B$, that is, for any sequences $u_n\in B$ and $t_n\to\infty$ the closure $[S(t_n)u_n]_\Phi$ is a compact set in $\Phi$. Then the following assertions are valid. 1) The set $\omega(B)$ is not empty. 2) It attracts the images of the set $B$ in the following sense: for every neighbourhood $\mathcal O(\omega(B))$, there exists a moment of time $T=T(\mathcal O)$ such that
$$
\begin{equation}
S(t)B\subset\mathcal O(\omega(B)),\qquad t\geqslant T.
\end{equation}
\tag{2.4}
$$
Assume in addition that the operators $S(t)$ are continuous for every fixed $t$. Then the $\omega$-limit set is strictly invariant: This lemma, which is proved in the next section, is the main building block of attractor theory, and all versions of attractors (known to the author) use the formula for an $\omega$-limit set implicitly or explicitly. We also emphasize here that the concept of an $\omega$-limit set requires asymptotic compactness, so if we want to have an attractor in our phase space, we need to fix the topology in $\Phi$ in such a way that the assumption of asymptotic compactness holds. The situation is much simpler in the finite-dimensional case $\Phi=\mathbb{R}^n$, where the dissipative estimate:
$$
\begin{equation}
\|S(t)u_0\|_{\Phi}\leqslant Q(\|u_0\|_\Phi)e^{-\alpha t}+ C_*
\end{equation}
\tag{2.6}
$$
(where $\alpha>0$, $C_*>0$, and $Q$ is some monotone increasing function) is enough to obtain asymptotic compactness. However, this estimate is not enough in infinite-dimensional case, so an extra work is required in order to verify this compactness. The next principal question is what objects should be attracted by the attractor? There are different answers to this question. First of all, there are local attractors which are very natural for the modern theory of dynamical systems (see, for example, [116]) and which attract only the trajectories starting from some (small) neighbourhood of the attractor. A more advanced version is a Milnor attractor, where the attraction property holds for the initial data up to a zero-measure set: see [176], see also [48], [88], [100], [108], [202] for more delicate versions of attractors. However, some global versions of attractors are traditionally preferable in the theory of dissipative PDEs, which is partially explained by the ultimate goal of this theory, to justify finite-dimensional reduction. The most widespread is a global attractor, which attracts the images of all bounded sets (see [12], [93], [209], [215]). Definition 2.3. A set $\mathcal A$ is a global attractor for the dynamical system $S(t)\colon\Phi\to\Phi$ if 1) the set $\mathcal A$ is compact in $\Phi$; 2) it is strictly invariant, that is, $S(t)\mathcal A=\mathcal A$ for all $t\geqslant 0$; 3) the set $\mathcal A$ is an attracting set for the semigroup $S(t)$, that is, for any bounded $B\subset \Phi$ and any neighbourhood $\mathcal O(\mathcal A)$ of the set $\mathcal A$, there exists $T=T(\mathcal O,B)$ such that for all $t\geqslant T$.Thus, on the one hand a global attractor consists of the complete trajectories of a dynamical system and contains all of the non-trivial dynamics (due to the attraction property). On the other hand, due to the compactness property, it is essentially smaller than the initial (usually infinite-dimensional) phase space, so the existence of a global attractor gives us already some kind of reduction of the degrees of freedom for the limit dynamics of the dissipative system under consideration. The second possibility is a point attractor, which attracts single trajectories only (not bounded sets of trajectories). Another possibility is a $(\Psi,\Phi)$-attractor, which was introduced in [12] and which attracts bounded sets in the space $\Psi$ in the topology of the space $\Phi$. It may also be natural to consider the attraction of only compact sets of initial data, and so on. The unified approach to these attractors will be considered in the next section, and here we state only the simplest and most popular version of the existence theorem for attractors (see [12], [215] for more details). Theorem 2.4. Let the operators $S(t)$ be continuous for every fixed $t\geqslant 0$, and let the semigroup $S(t)$ possess a compact attracting set $\mathcal B$ in $\Phi$, that is, for every bounded set $B$ and every neighbourhood $\mathcal O(\mathcal B)$ there exists $T=T(\mathcal O,B)$ such that Then the semigroup $S(t)$ possesses a global attractor $\mathcal A$, which is a subset of $\mathcal B$. This theorem is an almost immediate corollary of the properties of $\omega$-limit sets formulated in Lemma 2.2. Indeed, the desired attractor can be found via the following two equivalent formulae:
$$
\begin{equation*}
\mathcal A=\omega(\mathcal B)=\biggl[\,\bigcup_{B\text{ is bounded}} \omega(B)\biggr]_{\Phi}.
\end{equation*}
\notag
$$
However, the second expression is more general and works also for other types of attractors, without the continuity assumption, and so on. Similar theorems also hold for other types of attractors; one just needs to understand the attracting set in the proper way, for example, as point-attracting, $(\Phi,\Psi)$-attracting, and so on. For instance, a point attractor can be found via
$$
\begin{equation*}
\mathcal A_{\rm point}= \biggl[\,\bigcup_{u_0\in\Phi}\omega(u_0)\biggr]_{\Phi}.
\end{equation*}
\notag
$$
We also mention an important property of global attractors, namely, a representation formula: where $\mathcal K$ is the set of all complete bounded orbits of the semigroup $S(t)$:
$$
\begin{equation*}
\mathcal K:=\Bigl\{u:\mathbb{R}\to\Phi\mid S(t)u(h)=u(t+h), \ t\in\mathbb{R}_+,\ h\in\mathbb{R},\ \sup_{t\in\mathbb{R}}\|u(t)\|_{\Phi}\leqslant C_u\Bigr\}.
\end{equation*}
\notag
$$
Formula (2.7) is actually one of the main technical tools to work with global attractors. In particular, as we will see below, it is extremely useful in perturbation theory, in estimates for the Hausdorff and fractal dimensions, and so on. However, it may be not true for different types of attractors; for example, it fails in general for point attractors or in the case where the operators $S(t)$ are not continuous. We defer the general theory of attractors to the next sections and turn to examples which illustrate their basic properties. Example 2.5. Let us start with the first-order scalar ODE Example 2.6. More interesting things may happen when we consider equations on the plane. To plot the phase portraits on a plane is more transparent than to write down the explicit equations generating the corresponding dynamics, so we describe below the dynamics by plotting the corresponding pictures. In this example we consider two dynamical systems (see Fig. 1). These systems have a saddle point at the origin and two foci, say at $(1,1)$ and $(-1,-1)$. In Fig. 1, (a), the foci are stable, so all trajectories converge to one of these equilibria. Thus, $\mathcal A_{\rm point}$ consists of these equilibria, and the global attractor $\mathcal A_{\rm gl}$ contains these equilibria together with two unstable separatrices of the saddle shown by red lines in the picture. Thus, topologically, the global attractor is still a line segment, and both the Hausdorff and fractal dimensions of it are equal to one. In Fig. 1, (b), the foci become unstable via the Andronov–Hopf bifurcation and two extra limit cycles are born (see, for example, [210]). In this case the point attractor consists of three equilibria together with the newly born limit cycles. Now the global attractor contains two extra discs bounded by limit cycles which are shown by blue lines in the picture (together with the equilibria, limit cycles, and two unstable separatrices of the saddles, shown by red colour). This follows, for example, from the fact that the global attractor consists of all complete bounded orbits. Both the Hausdorff and fractal dimensions are equal to two in this case. Remarkable here is that the global attractor remains connected, but becomes not linearly connected after the bifurcation. It is also interesting to look at the instant of bifurcation. At this moment, the phase portrait remains topologically the same as in Fig. 1, (a), but the foci become degenerate. In a generic case the radius $R$ of the spirals satisfies $\dfrac{dR}{dt}\sim R^3$, so $R\sim t^{-1/2}\sim\varphi^{-1/2}$, where $\varphi$ is an angle of the spiral. An elementary computation shows that the fractal dimension of such a spiral is $4/3$. Thus, since the unstable separatrices of the saddle form such spirals, we have
$$
\begin{equation}
\dim_{\rm f}(\mathcal A_{\rm gl})=\frac{4}{3}\ne \dim_{\rm H}(\mathcal A_{\rm gl})=1
\end{equation}
\tag{2.8}
$$
(the Hausdorff dimension is one since the attractor is a countable union of 1D segments). This is the simplest mechanism which generates global attractors with non-integer fractal dimension, as well as with inequal Hausdorff and fractal dimensions. We also mention that the Lyapunov dimension of the attractor at the instant of bifurcation is obviously equal to two. This phase portrait consists of a saddle-node at the origin $(0,0)$ glued with the disc $\{(x,y)\in\mathbb{R}^2\colon x^2+(y-1)^2\leqslant 1\}$ filled by homoclinic orbits to the origin. The key feature of this dynamical system is that the $\omega$-limit set of any single trajectory coincides with the origin, and therefore the point attractor $\mathcal A_{\rm point}=\{0\}$ is trivial, but the global attractor
$$
\begin{equation*}
\mathcal A_{\rm gl}=\{(x,y)\in\mathbb{R}^2\colon x^2+(y-1)^2\leqslant1\}
\end{equation*}
\notag
$$
is not trivial. Note that this phase portrait is extremely degenerate and an arbitrarily small perturbation will destroy homoclinic orbits and produce a “big” limit cycle. Thus, in contrast to the global attractor, the point attractor is in general not upper semicontinuous with respect to perturbations, and this is one more reason why the global attractor looks preferable. Example 2.8. The situation becomes much more interesting when the dimension of the phase space is larger than two, since we do not have the Poincaré–Bendixon theorem any longer and more complicated structures than equilibria, limit cycles and homo/heteroclinic connections between them may appear. One of the most popular 3D model examples here is the Lorenz attractor (see [156]), which illustrates the possibility of a chaotic “unpredictable” behaviour in deterministic systems, the so-called deterministic chaos. It is strongly believed that similar effects are responsible for the complicated behaviour of more realistic systems (including PDEs) arising, for example, in hydrodynamics, weather prediction, chemical reactions, and so on. The Lorenz system consists of the following equations:
$$
\begin{equation}
\frac{dx}{dt}=\sigma(y-x),\qquad \frac{dy}{dt}=x(\rho-z)-y,\qquad \frac{dz}{dt}=xy-\beta z
\end{equation}
\tag{2.9}
$$
and the standard choice of the parameters is $\sigma=10$, $\beta=8/3$, and $\rho=28$. The corresponding attractor is shown in Fig. 3 (the attractor may be essentially different for other values of the parameters). The structure of the Lorenz attractor is well understood nowadays, but this theory is far beyond the scope of this survey, so we restrict ourselves to mentioning a few of interesting properties of this object. Further details can be found in [1], [87], [91], [92], [103], [118], [15], [189]. It is known that the Lorenz system is dissipative and possesses an absorbing ellipsoid $\mathcal B\subset \mathbb{R}^3$, which is semi-invariant (see, for example, [215]) and, therefore the global attractor $\mathcal A_{\rm gl}$ of (2.9) exists. It is a fractal set whose Hausdorff and fractal dimensions are strictly between 2 and 3:
$$
\begin{equation*}
2<\dim_{\rm H}(\mathcal A)\leqslant\dim_{\rm f}(\mathcal A)<3
\end{equation*}
\notag
$$
(at least for the standard values of the parameters; see [183]). In particular, the upper bounds follow from the volume contraction method and the explicit formula for the Lyapunov dimension of this attractor:
$$
\begin{equation*}
\dim_{\rm L}(\mathcal A)=3-\frac{2(\sigma+\beta+1)}{\sigma+1+ \sqrt{(\sigma-1)^{2}+4\sigma\rho}}\,,
\end{equation*}
\notag
$$
which holds for all values of the parameters for which all three equilibria are hyperbolic (see [151]). It is worth emphasizing that this global attractor $\mathcal A_{\rm gl}$ does not coincide with the classical Lorenz attractor $\mathcal A_{\rm lor}$ or geometric Lorenz attractor, whose existence was established by Tucker using interval arithmetics and the computer assistant proof (see [218]). Namely, he verified the existence of a compact semi-invariant domain $U$ in $\mathbb{R}^3$ (inside of the absorbing ellipsoid $\mathcal B$). This domain looks like a solid double torus which contains the zero saddle equilibrium $C_0$, but does not contain the other two saddle-foci $C_\pm$. The local attractor of this domain $U$ is exactly the Lorenz attractor $\mathcal A_{\rm lor}$, which possesses a geometric description, proposed in [1], [91], in terms of iterations of a 1D discontinuous Poincaré map, and the main achievement of [218] was the rigorous numerical verification of the fact that the Lorenz system with the classical values of the parameters satisfies all the hypotheses stated in the geometric model. In particular, it follows from the assumptions verified that $\mathcal A_{\rm lor}$ is topologically transitive, pseudo-hyperbolic, possesses an invariant measure with nice properties (a Sinai–Ruelle–Bowen measure), and the periodic orbits are dense in it. On the other hand there is a strong numerical evidence that there are no complete bounded trajectories belonging to the domain $\mathcal B\setminus U$ in the Lorenz system with the standard parameters, although, to the best of our knowledge, this has not been verified rigorously yet on the level of computer assistant proofs. If we believe in this fact, then the relations between global, Lorenz, and point attractors for the Lorenz system are given by
$$
\begin{equation*}
\mathcal A_{\rm point}=\mathcal A_{\rm lor}\cup C_+\cup C_-,\qquad \mathcal A_{\rm gl}=\mathcal A_{\rm lor}\cup \mathcal M^+(C_+)\cup \mathcal M^+(C_-),
\end{equation*}
\notag
$$
where $\mathcal M^+(C_\pm)$ are the two-dimensional unstable manifolds of the saddle-foci $C_\pm$ and all unions are disjoint. Example 2.9. We now turn to examples of PDEs and start with the heat equation in $\mathbb{R}^d$: There are several possibilities to extend the theory of global attractors to this case based on different classes of “bounded” sets which must be attracted to it or/and different topologies of the phase space. First of all, we may restrict our attention to one-point sets, then we already have the attractor. However, doing that, we lose a lot of information (clearly the attraction is actually stronger). The second possibility would be to restrict the attraction property to compact sets only. This is probably the most appropriate choice for equation (2.10) since we have indeed the uniformity of attraction for compact sets only. Other possibilities are related to the change of the topology of attraction. Indeed, the attraction of all bounded sets of $\Phi=L^2(\mathbb{R}^d)$ holds if we endow the phase space $\Phi$ with the topology induced by the embedding $\Phi\subset \Phi_{\rm loc}:=L^2_{\rm loc}(\mathbb{R}^d)$ (again, this is a straightforward corollary of the explicit formula for solutions). This corresponds to the locally compact attractor, which attracts bounded sets in the local topology only and which is typical for PDEs in unbounded domains (see [179] and the references therein for more details). Alternatively, we may endow the space $\Phi=L^2(\mathbb{R}^d)$ with the weak topology. Then all bounded sets are attracted in this topology. In all these cases the attractor consists of the single point $\{0\}$. Example 2.10. We complete this section by the model example of a nonlinear heat equation in a bounded domain, which demonstrates the standard way how the abstract result of Theorem 2.4 is used to verify the existence of an attractor for nonlinear PDEs. Namely, we consider the following problem: Indeed, multiplying (2.11) by $u$ and integrating with respect to $x$ we have
$$
\begin{equation*}
\frac{1}{2}\,\frac{d}{dt}\|u(t)\|^2_{L^2}+\|\partial_x u(t)\|_{L^2}^2+ \|u(t)\|^4_{L^4}=a\|u(t)\|^2_{L^2}.
\end{equation*}
\notag
$$
Using Poincaré’s inequality $\|\partial_x u\|^2_{L^2}\geqslant \alpha\|u\|^2_{L^2}$, together with the obvious estimate $a\|u\|^2_{L^2}\leqslant \|u\|^4_{L^4}+Ca^2$, we arrive at
$$
\begin{equation}
\frac{d}{dt}\|u(t)\|^2_{L^2}+\alpha\|u(t)\|^2_{L^2}+ \|\nabla_x u(t)\|^2_{L^2}\leqslant Ca^2,
\end{equation}
\tag{2.12}
$$
where $C$ is a constant independent of $a$, and, after integration with respect to time, we obtain the desired dissipative estimate in $\Phi$:
$$
\begin{equation}
\|u(t)\|^2_{L^2}\leqslant \|u(0)\|^2_{L^2}\,e^{-\alpha t}+C_*a^2,
\end{equation}
\tag{2.13}
$$
where $C_*=C/\alpha$. The global existence of a solution is straightforward here and can be verified, for example, by using the Galerkin approximations (see, for instance, [12], [215] for more details). Let us verify uniqueness. Indeed, let $u_1(t)$ and $u_2(t)$ be two solutions of (2.11), and let $v(t)=u_1(t)-u_2(t)$. Then this function solves
$$
\begin{equation}
\partial_t v-\partial_x^2 v=av-(u_1^2+u_1u_2+u_2^2)v, \qquad v\big|_{t=0}=u_1(0)-u_2(0).
\end{equation}
\tag{2.14}
$$
Multiplying this equation by $v(t)$ and using that $u_1^2+u_1u_2+u_2^2\geqslant 0$, we have
$$
\begin{equation*}
\frac{d}{dt}\|v(t)\|^2_{L^2}+2\|\partial_x v(t)\|^2_{L^2}\leqslant 2a\|v(t)\|^2_{L^2}
\end{equation*}
\notag
$$
and
$$
\begin{equation}
\|u_1(t)-u_2(t)\|^2_{L^2}+2\int_0^t e^{2a(t-s)}\|\partial_x v(s)\|^2_{L^2}\,ds \leqslant e^{2at}\|u_1(0)-u_2(0)\|^2_{L^2},
\end{equation}
\tag{2.15}
$$
which proves uniqueness. Thus, we have constructed a solution semigroup $S(t)\colon\Phi\to\Phi$ associated with equation (2.11). This semigroup possesses a dissipative estimate (2.6) in the phase space $\Phi=L^2(\Omega)$ and, for this reason, the set
$$
\begin{equation*}
\mathcal B:=\{u_0\in\Phi\colon \|u_0\|^2_{\Phi}\leqslant 2C_*a^2\}
\end{equation*}
\notag
$$
is a bounded attracting (and even absorbing) set for this semigroup. However, this is still not enough to verify the existence of a global attractor since the set $\mathcal B$ is not compact in $\Phi$. To overcome this difficulty we must use the parabolic smoothing property. To obtain the desired smoothing property, we multiply (2.11) by $t\,\partial_x^2 u$ and integrate with respect to $x$. This gives
$$
\begin{equation*}
\begin{aligned} \, \frac{1}{2}\,\frac{d}{dt}\bigl(t\|\partial_x u(t)\|^2_{L^2}\bigr)&= -t\|\partial^2_x u(t)\|^2_{L^2}-3t\|u\partial_x u\|^2_{L^2}+ \biggl(at+\frac{1}{2}\biggr)\|\partial_x u(t)\|^2_{L^2} \\ &\leqslant \biggl(at+\frac{1}{2}\biggr)\|\partial_x u(t)\|^2_{L^2}. \end{aligned}
\end{equation*}
\notag
$$
Integrating this estimate with respect to $t\in[0,1]$, we arrive at
$$
\begin{equation*}
\|\partial_x u(1)\|^2_{L^2}\leqslant (2a+1)\int_0^1\|\partial_x u(s)\|^2_{L^2}\,ds.
\end{equation*}
\notag
$$
Finally, we estimate the right-hand side of this inequality by integrating estimate (2.12) with respect to $t\in[0,1]$ and using the dissipative estimate (2.13). This gives
$$
\begin{equation*}
\|\partial_x u(1)\|_{L^2}^2\leqslant C(a+1)\|u(0)\|^2_{L^2},
\end{equation*}
\notag
$$
where the constant $C$ is independent of the parameter $a$. Shifting time, we also see that
$$
\begin{equation*}
\|\partial_x u(t+1)\|_{L^2}^2\leqslant C(a+1)\|u(t)\|^2_{L^2}.
\end{equation*}
\notag
$$
Combining this estimate with (2.13), we see that the set
$$
\begin{equation}
\mathcal B_1:=\bigl\{u_0\in H^1_0(\Omega)\colon\|\partial_x u_0\|_{L^2}^2 \leqslant 2CC_*a^2(a+1)\bigr\}
\end{equation}
\tag{2.16}
$$
is also an attracting (and even absorbing) set for the semigroup $S(t)$. The difference is that this set is compact in $\Phi$. The continuity of the operators $S(t)$ with respect to the initial data follows from estimate (2.15). Thus, all the assumptions of Theorem 2.4 are verified and the existence of a global attractor $\mathcal A_{\rm gl}\subset\Phi$ for (2.11) is established. The next question is the structure of the global attractor $\mathcal A_{\rm gl}$. The key role in its further investigation is played by the global Lyapunov function. Indeed, multiplying (2.11) by $\partial_t u$ and integrating with respect to $x$ we derive that
$$
\begin{equation*}
\frac{d}{dt}\mathcal L(u(t)):=\frac{d}{dt}\biggl(\frac{1}{2} \|\partial_x u(t)\|^2_{L^2}+\frac{1}{4}\|u(t)\|^4_{L^4}- \frac{a}{2}\|u(t)\|^2_{L^2}\biggr)=-\|\partial_t u(t)\|^2_{L^2}
\end{equation*}
\notag
$$
and see that the function $u\to\mathcal L(u)$ is strictly decreasing along non-equilibrium trajectories. This allows us to conclude that any trajectory of (2.11) tends to the set $\mathcal R$ of equilibria as $t\to\infty$. The equilibria $u_0\in\mathcal R$ solve a second-order scalar ODE, and their structure can completely be understood. In particular, it is known that new equilibria can bifurcate from the zero equilibrium only, and all non-zero equilibria remain hyperbolic for all values of the parameter $a$. This fact shows that there are exactly $N=2\lceil \sqrt a\,\rceil-1$ different equilibria in $\mathcal A_{\rm gl}$ for $a>1$ (the case $a<1$ is not interesting since then we have a single globally exponentially stable zero equilibrium). Thus, from this fact and the existence of a global Lyapunov function we conclude that any trajectory of the equation under consideration stabilizes, as $t\to\infty$, to one of these equilibria:
$$
\begin{equation*}
\mathcal A_{\rm point}=\mathcal R=\{u_1,u_2,\dots,u_N\}
\end{equation*}
\notag
$$
and the global attractor $\mathcal A_{\rm gl}$ consists of the equilibria and heteroclinic orbits connecting them. A more detailed analysis based on the Morse–Smale theory allows us to show the robustness of the dynamics on the attractor for $a\in(n^2,(n+1)^2)$, $n\in\mathbb N$, as well as to determine what equilibria are connected by heteroclinic orbits and to compute the dimensions of the corresponding sets of heteroclinic orbits (see [12], [73], [97] for more details). To conclude we note that (2.11) is a rare exception in attractor theory, where the structure of the global attractor can be understood fully. This is possible thanks to three different nice properties of this equation. The first is the existence of the global Lyapunov function $\mathcal L$ discussed above. The second is the existence of another type of (discrete) Lyapunov function, namely, the number $Z(u(t))$ of zeros of the profile $x\to u(t,x)$ which is a non-increasing function of time. Exactly this discrete Lyapunov function is responsible for the transversality of the stable and unstable manifolds of equilibria and the Morse–Smale property. And the third property is that the equation under consideration is order preserving with respect to the cone of non-negative functions. This allows us to apply the Perron–Frobenius theory and simplify essentially the analysis of the spectral properties of equilibria (see [12], [29], [97], [138] for more details). In contrast to this, the structure of the global attractor remains a mystery in a more or less general situation (say, for the Navier–Stokes equations) and our knowledge is limited to the general facts that it is compact, connected, consists of bounded trajectories, and has finite Hausdorff and fractal dimensions. 3. Attractors: a unified approachIn this section we present a unified approach to various types of attractors. As we have seen in the previous section, we need to specify two major things: 1) bornology: what sets will be attracted by the attractor (we will call such sets “bounded” which explains the denomination bornology); 2) topology: in what sense attraction will hold. We assume here that our phase space $\Phi$ is a Hausdorff topological space. On the one hand, the main results of the theory discussed in the previous section can easily be extended to general Hausdorff topological spaces and, on the other hand, this extension makes the theory more convenient and more elegant, especially when attraction in a weak or weak-star topology is considered. Then we fix a family of sets $\mathbb B\subset 2^{\Phi}$, which will be referred to as a bornology (or a bounded structure) on the phase space $\Phi$. Recall that usually the definition of a bounded structure includes some assumptions on $\mathbb B$, such as stability with respect to inclusions and finite unions, as well as the fact that $\Phi=\bigcup\limits_{B\in\mathbb B}B$ (see, for example, [102]). Since the attraction of a set $B$ automatically implies the attraction of all of its subsets and the same is true for finite unions, the first two assumptions are actually not restrictive (we can always extend our bornology to satisfy them without changing the attraction properties). However, the third assumption is a big restriction since it excludes local attractors, so we prefer not to impose it. We also do not impose the first two assumptions since they can be satisfied by the straightforward extension mentioned above, so, in our theory, $\mathbb B$ is an arbitrary family of non-empty sets. Finally, we are given a semigroup $S(t)\colon\Phi\to\Phi$ acting on our phase space, which is not assumed to be continuous, so we need some modifications in the theory. We start with the standard definitions. Definition 3.1. A set $\mathcal B\subset\Phi$ is an absorbing ($\mathbb B$-absorbing) set for the semigroup $S(t)$ if, for every $B\in\mathbb B$, there exists time $T=T(B)$ such that A set $\mathcal B$ is an attracting ($\mathbb B$-attracting) set for the semigroup $S(t)$ if, for every $B\in\mathbb B$ and every neighbourhood $\mathcal O(\mathcal B)$ of $\mathcal B$, there exists $T=T(B,\mathcal O)$ such that We now turn to attractors. The key difference here is that the invariance of an attractor $\mathcal A$ requires some kind of continuity of $S(t)$ which we do not assume, so we need to replace invariance by minimality. Definition 3.2. A set $\mathcal A\subset\Phi$ is an attractor (a $\mathbb B$-attractor) of a semigroup $S(t)$ acting on the topological space $\Phi$ if the following assumptions are satisfied: 1) $\mathcal A$ is a compact set in $\Phi$; 2) $\mathcal A$ is an attracting set (a $\mathbb B$-attracting set) for the semigroup $S(t)$; 3) $\mathcal A$ is a minimal set (with respect to inclusion) which satisfies properties 1) and 2). We are now ready to state the main result of this section, which is related to the existence of an attractor. Theorem 3.3. Let $S(t)\colon\Phi\to\Phi$ be a semigroup acting on a Hausdorff topological space $\Phi$ endowed with a bornology $\mathbb B$. Assume that $S(t)$ possesses a compact $\mathbb B$-attracting set $\mathcal B$. Then there exists a $\mathbb B$-attractor $\mathcal A$, which can be obtained as follows: Proof. Although the proof looks more or less standard and various particular cases of this theorem can be found in the literature (see [38], [170]), for the convenience of the reader we give all details here. As usual, we need to verify the analogue of (2.2) for our case.
Step 1: the set $\omega(B)$ is not empty for all non-empty $B\in\mathbb B$. Let
$$
\begin{equation*}
K_T:=\biggl[\,\bigcup_{t\geqslant T}S(t)B\biggr]_{\Phi},\quad U_T=\Phi\setminus K_T.
\end{equation*}
\notag
$$
Assume that $\omega(B)=\bigcap\limits_{T\geqslant 0}K_T=\varnothing$. Then $\{U_T\}_{T\geqslant 0}$, is an open covering of $\Phi$ and, in particular, an open covering of the compact set $\mathcal B$. Since the $U_T$ are non-empty and nested, there exists $T\geqslant 0$ such that $\mathcal B\subset U_T$, and therefore
$$
\begin{equation*}
K_T\cap\mathcal B=\biggl[\,\bigcup_{t\geqslant T}S(t)B\biggr]_{\Phi}\cap \mathcal B=\varnothing.
\end{equation*}
\notag
$$
We claim that this contradicts the attraction property. Indeed, let $x\in\mathcal B$ be arbitrary. Then there exists a neighbourhood $U_x$ of $x$ such that $U_x\cap \bigcup\limits_{t\geqslant T}S(t)B=\varnothing$. Let $U=\bigcup\limits_{x\in\mathcal B}U_x$. Then on the one hand $U$ is a neighbourhood of $\mathcal B$ and on the other Therefore, $U\cap S(t)B=\varnothing$ for all $t\geqslant T$ and $\mathcal B$ is not an attracting set for $B$. This contradiction proves that $\omega(B)$ is not empty.
Step 2: $\omega(B)\subset\mathcal B$. Indeed, let $x\in\omega(B)\setminus\mathcal B$. Then, since $\Phi$ is Hausdorff and $\mathcal B$ is compact, there exist neighbourhoods $U_{\mathcal B}$ and $U_x$ of the set $\mathcal B$ and the point $x$, respectively, such that $U_{\mathcal B}\cap U_x=\varnothing$. Then we know from the attraction property that $S(t)B\subset U_{\mathcal B}$ for all $t\geqslant T$. On the other hand we know that $x\in K_T$ for all $T\geqslant 0$. From this we conclude that $\bigcup\limits_{t\geqslant T}S(t)B\cap U_x\ne\varnothing$ for all $T\geqslant 0$. This contradicts the fact that $U_{\mathcal B}\cap U_x=\varnothing$ and proves the statement. Step 3: $\omega(B)$ attracts $B$. Note that $\omega(B)$ is compact as a closed subset of a compact set. Let $U$ be an arbitrary neighbourhood of $\omega(B)$. Consider the closed sets Then $\bigcap\limits_{T\geqslant 0}C_T= \bigl(\bigcap\limits_{T\geqslant 0}K_T\bigr)\setminus U=\varnothing$ (since $\bigcap\limits_{T\geqslant 0}K_T=\omega(B)\subset U$). Thus, the sets $V_T:=\Phi\setminus C_T$ cover the compact set $\mathcal B$. Extracting a finite covering and using the fact that the sets $V_T$ are increasing, we get that $\mathcal B\subset V_T$ for some $T$. Thus, $C_T\cap B=\varnothing$. Since $C_T$ is closed and $\mathcal B$ is compact, there is a neighbourhood $U_{\mathcal B}$ of $\mathcal B$ such that $C_T\cap U_{\mathcal B}=\varnothing$. In particular, $(S(t)B\setminus U)\cap U_{\mathcal B}=\varnothing$ for all $t\geqslant T$ (since $S(t)B\subset K_T$ for $t\geqslant T$). Since $S(t)B\subset U_{\mathcal B}$ for a sufficiently large $t$ due to the attraction property, we must have $S(t)B\subset U$ if $t$ is large enough, which proves the attraction property.Step 4: the attractor. We define it by formula (3.1). Then, since $\omega(B)\subset\mathcal B$ for all $B\in\mathbb B$, $\mathcal A$ is a closed subset of the compact set $\mathcal B$, so it is compact in $\Phi$. The attraction property is also obvious since, by definition, $\mathcal A$ contains $\omega(B)$ for every $B\in\mathbb B$. Thus, we only need to check minimality. To this end, it is enough to prove that $\omega(B)$ is a minimal set which attracts $B$. Indeed, let $\Omega(B)$ be another compact attracting set for $B$. Then, arguing as at Step 2, we get that $\omega(B)\subset\Omega(B)$ and $\omega(B)$ is indeed minimal. This finishes the proof of Theorem 3.3. Let us now discuss the invariance of the attractor $\mathcal A$. This requires some kind of continuity of the operators $S(t)$. Proposition 3.4. Let the assumptions of Theorem 3.3 hold, and let, in addition, the operators $S(t)\colon\Phi\to\Phi$ be continuous for every fixed $t$. Then the attractor $\mathcal A$ is strictly invariant with respect to $S(t)$, that is, for all $t\geqslant 0$. Proof. First we prove that $\mathcal A\subset S(t)\mathcal A$. Indeed, $S(t)\mathcal A$ is compact as a continuous image of a compact set. Let us prove that $S(t)\mathcal A$ is an attracting set. Let $U$ be a neighbourhood of $S(t)\mathcal A$. Then, by continuity, $V:=S(t)^{-1}(U)$ is a neighbourhood of $\mathcal A$. Due to the attraction property for $\mathcal A$, for any $B\in\mathbb B$ there exists time $T=T(B,V)$ such that $S(h)B\subset V$ for all $h\geqslant T$. Then $S(h)B\subset U$ for all $h\geqslant T+t$ and the attraction property is proved. Thus, by the minimality of $\mathcal A$ we have the desired inclusion.
Let us prove the reverse inclusion. Indeed, for any $B\in\mathbb B$, we have
$$
\begin{equation*}
\begin{aligned} \, S(t)\omega(B)&=S(t)\bigcap_{T\geqslant 0}\biggl[\,\bigcup_{h\geqslant T} S(h)B\biggr]_{\Phi}\subset \bigcap_{T\geqslant 0} S(t)\biggl[\,\bigcup_{h\geqslant T}S(h)B\biggr]_{\Phi} \\ &\subset \bigcap_{T\geqslant 0}\biggl[S(t)\bigcup_{h\geqslant T} S(h)B\biggr]_{\Phi}=\bigcap_{T\geqslant 0}\biggl[\,\bigcup_{h\geqslant T+t} S(h)B\biggr]_{\Phi}=\omega(B). \end{aligned}
\end{equation*}
\notag
$$
Using again that $S(t)[A]_\Phi\subset [S(t)A]_\Phi$ due to continuity, in combination with (3.1), we conclude that $S(t)\mathcal A\subset\mathcal A$ and finish the proof of the proposition. The next example shows that the continuity assumption cannot be omitted. Example 3.5. Let $\Phi:=(-\infty,0]\cup\{1\}$ with the standard topology and bornology induced by the embedding in $\mathbb{R}$, and let Then, obviously, $\mathcal A=\{0,1\}$. However, $\omega(\mathcal A)=\{1\}\ne\mathcal A$ and $S(t)\mathcal A=\{1\}\ne \mathcal A$. This example shows that we cannot expect the strict invariance of the attractor without continuity assumptions. Moreover, if $\mathcal B$ is a compact attracting set of $S(t)$, then in general we cannot expect that Indeed, in our case we may take $\mathcal B=\mathcal A$. In addition, this example shows that the condition that the $S(t)\colon\mathcal B\to \Phi$ are continuous is not enough to have invariance (again, the $S(t)$ are continuous on $\mathcal B:=\mathcal A$ in our example) and we really need to assume continuity on a larger set than $\mathcal B$. We now discuss the validity of the representation formula (2.7). We define a bounded complete trajectory $u\colon\mathbb{R}\to\Phi$ as a full trajectory of $S(t)$ such that and we consider the set $\mathcal K$ of all complete bounded trajectories with respect to a given bornology $\mathbb B$.It is immediate to see from the attraction property that any complete bounded trajectory belongs to the attractor $\mathcal A$, so we have However, the opposite embedding is not true in general and requires some extra assumptions. In particular, we need some kind of continuity in order to get the invariance of the attractor (see Example 3.5, where there are no complete bounded trajectories passing through the point $0\in\mathcal A$). This continuity is still not enough to get the representation formula (2.7). For instance, if the bornology $\mathbb B$ consists of one-point sets only, then complete bounded trajectories are equilibria only, so (2.7) holds for the corresponding point attractor $\mathcal A_{\rm point}$ if and only if it coincides with the set of equilibria ($\mathcal A_{\rm point}=\mathcal R$). The next proposition gives a useful sufficient condition for the validity of (2.7).Proposition 3.6. Let $\Phi$ be a Hausdorff topological space with a bornology $\mathbb B$ which is stable with respect to inclusions. Also let $S(t)\colon\Phi\to\Phi$ be a continuous semigroup in $\Phi$ (that is, all maps $S(t)$, $t\geqslant 0$, are continuous) which possesses a compact bounded attracting set $\mathcal B$. Then this semigroup possesses a $\mathbb B$-attractor $\mathcal A\subset\mathcal B$, which is strictly invariant and is generated by all complete bounded trajectories (that is, (2.7) holds). Proof. Indeed, the existence of the attractor $\mathcal A$ follows from Theorem 3.3. Its strict invariance was proved in Proposition 3.4, the embedding (3.3) follows from the attraction property, so we only need to verify that every point $u_0\in\mathcal A$ belongs to some complete bounded trajectory. This trajectory can be constructed as follows: for $t\geqslant 0$, we just define $u(t):=S(t)u_0\in\mathcal A$. By the invariance established, we have $u(t)\in\mathcal A$ for all $t\geqslant 0$. Let us now define $u(-n)$ for $n\in\mathbb N$ using induction. Indeed, by the strict invariance, there exists $u(-1)\in\mathcal A$ such that $S(1)u(-1)=u_0$; then there exists $u(-2)\in\mathcal A$ such that $S(1)u(-2)=u(-1)$, and so on. Finally, for $t\in[-n,-n+1]$, we set $u(t):=S(t+n)u(-n)$. Then, by invariance $u(t)\in\mathcal A$ for all $t\in\mathbb{R}$ and by construction $u(0)=u_0$. Since $B_u\in\mathcal A$ and $\mathcal A$ is bounded, using the stability of $\mathbb B$ with respect to inclusion, we conclude that $B_u\in\mathbb B$ and $u\in\mathcal K$. This proves the proposition. Remark 3.7. As we have seen, the continuity assumption for the semigroup $S(t)$ cannot be omitted. However, there are several possibilities to weaken it. One of them is to assume that the graph of the map $S(t)$ is closed in $\Phi\times\Phi$ for every fixed $t\geqslant 0$ (see [12], [185]). Another possibility is to verify this continuity not in the whole phase space $\Phi$, but on the absorbing set $B_0$ only. Namely, it is not difficult to verify that the strict invariance of the attractor remains true if $S(t)\colon B_0\to \Phi$ is continuous for every fixed $t$. This is especially useful in the case where $\Phi$ is endowed with a weak or weak-star topology. Although this topology is not metrizable on the whole space $\Phi$, its restriction to a properly chosen absorbing set is very often metrizable. This allows us to verify only the sequential continuity of the maps $S(t)$. This may be a great simplification since the topological continuity of the nonlinear maps $S(t)$ is much harder to check than sequential continuity (at least when we are speaking about weak or weak-star topologies). We also recall that in a general topological space compactness and sequential compactness are different (even unrelated), so the two corresponding types of attractors can be considered, topological and sequential ones, which may also be completely different. We will not go into the details here (see [137] for these details), but only mention two natural cases where they coincide. The first case is when we have a metrizable absorbing set and the second, more interesting case is when $\Phi$ is a Banach space endowed with the weak topology. Then compactness and sequential compactness coincide due to the Eberlein–Smulian theory. Remark 3.8. To conclude this section, we also mention that in many cases the key dissipative estimates give us the existence of an absorbing set $\mathcal B$, which is ‘bounded’ in $\Phi$ in a certain sense and for this reason is often a complete metrizable space even when the original phase space $\Phi$ is not metrizable. Then the verification of the existence of a compact attracting set is reduced to checking so-called asymptotic compactness, namely, we need to check that, for any sequences $t_n\to\infty$ and $u_n\in\mathcal B$, the sequence $\{S(t_n)u_n\}_{n=1}^\infty$ is precompact. Actually, it is not difficult to check that under the above assumptions this property is equivalent to the existence of a compact attracting set, but the use of asymptotic compactness arguments allows us to avoid constructing this compact attracting set explicitly, which is useful in many applications. At the moment there are many powerful methods for verifying asymptotic compactness, for example, ones based on the Kuratowski measure of non-compactness, or on the energy method, or on compensated compactness arguments. We refer the interested reader to [44], [93], [181] (see also the references therein) for more details. 4. Trajectory attractorsIn this section we demonstrate how attractor theory can be applied to equations without the uniqueness of solutions of the corresponding initial value problem. One possible approach to handle the non-uniqueness problem is to consider semigroups of multivalued maps and extend the concept of an attractor to such semigroups (see [11], [6], [13], [171]), however, there exists an alternative, more elegant approach which was developed in [36] and [208] for the study of the 3D Navier–Stokes equations and which we will use here (see also [225] for the case of elliptic equations). Under this approach, one constructs a trajectory dynamical system associated with the problem under consideration and then applies the usual theory of attractors to this dynamical system. We illustrate below the main idea by a number of examples. 4.1. ODEs and reaction-diffusion equationsWe start with considering a relatively simple case of ODEs and reaction-diffusion systems without uniqueness. Example 4.1. Consider a system of ODEs To construct an attractor for this semigroup we fix the standard bornology $\mathbb B$ in $\mathbb{R}^N$ which consists of usual bounded sets in $\mathbb{R}^N$. Then the ball
$$
\begin{equation*}
\mathcal B:=\biggl\{u\in\mathbb{R}^N\colon |u|^2\leqslant \frac{2C}{\alpha}\biggr\}
\end{equation*}
\notag
$$
is a compact bounded absorbing set for the semigroup $S(t)$. Moreover, the maps $S(t)$ are continuous with respect to the initial data. Thus, according to the general theory, there exists a global attractor $\mathcal A\subset\Phi$. To explain the idea of trajectory attractors, we consider the set $\mathcal K_+\subset C_{\rm loc}(\mathbb{R}_+,\Phi)$ which consists of all semi-trajectories $u(t)$, $t\geqslant 0$, of the solution semigroup $S(t)$ constructed that start from all $u_0\in\Phi$. Then the semigroup of time shifts $T(h)$, $h\geqslant 0$, acts on this set:
$$
\begin{equation}
(T(h)u)(t):=u(t+h),\quad t,h\in\mathbb{R}_+,\quad T(h)\mathcal K_+\subset \mathcal K_+.
\end{equation}
\tag{4.4}
$$
Thus, we have defined a dynamical system $T(h)$ on $\mathcal K_+$. This dynamical system is called a trajectory dynamical system associated with equation (4.1), and the set $\mathcal K_+$ is referred to as the trajectory phase space. Note that in the case where the uniqueness theorem holds, the trajectory dynamical system constructed is topologically conjugate to the semigroup $(S(t),\Phi)$ acting on the usual phase space $\Phi$. Indeed, let $\mathbb S\colon\Phi\to \mathcal K_+$ be the solution operator $u_0\to u(\,\cdot\,)$ associated with equation (4.1). Then this map is one-to-one with the inverse map defined via $\mathbb S^{-1}u:=u(0)$, and it is a homeomorphism if we endow the set $\mathcal K_+$ with the topology induced by the embedding in $C_{\rm loc}(\mathbb{R}_+,\Phi)$. Moreover, we have the obvious relation which shows that the trajectory dynamical system $(T(h),\mathcal K_+)$ is equivalent to the classical one $(S(t),\Phi)$. Finally, if we define a bornology $\mathbb B_{\rm tr}$ on $\mathcal K_+$ by
$$
\begin{equation}
\mathbb B_{\rm tr}=\bigl\{B\subset\mathcal K_+\colon B\big|_{t=0} \text{ is bounded in } \Phi\bigr\},
\end{equation}
\tag{4.5}
$$
then we also have a one-to-one correspondence between the bounded sets in $\mathcal K_+$ and in $\Phi$. Thus, since $(S(t),\Phi)$ possesses a global attractor $\mathcal A$ (as explained above), the trajectory dynamical system $(T(h),\mathcal K_+)$ also possesses an attractor $\mathcal A_{\rm tr}:=\mathbb S\mathcal A$, which is called the trajectory attractor associated with equation (4.1). This attractor is strictly invariant with respect to the shift semigroup and consists of complete bounded trajectories: Note also that, due to the dissipative estimate (4.3), the bornology $\mathbb B_{\rm tr}$ consists of all bounded sets in $C_{\rm loc}(\mathbb{R},\Phi)\cap \mathcal K_+$ (or even $C_b(\mathbb{R},\Phi)\cap \mathcal K_+$). Up to the moment, the above constructions look as a tautology; however, the construction of the trajectory dynamical system $(T(h),\mathcal K_+)$ does not require the uniqueness theorem to hold, so it can be extended to cases where the uniqueness theorem is either violated or not known. For instance, in the example under consideration we may relax the assumption of the Lipschitz continuity of $f$ and assume that $f$ is only continuous, but satisfies the dissipativity assumption (4.2). Then we still have the existence of a solution for any $u_0\in\Phi$ and the validity of dissipative estimate (4.3), so we may construct the trajectory dynamical system $(T(h),\mathcal K_+)$ and the bornology $\mathbb B_{\rm tr}$ in the same way as before, and we may speak about the (trajectory) attractor $\mathcal A_{\rm tr}$ of this system despite the fact that the uniqueness theorem does not hold here in general and the classical dynamical system $(S(t),\Phi)$ is not even properly defined. Proposition 4.2. Let the function $f$ be continuous and satisfy the dissipativity assumption (4.2). Then the trajectory dynamical system $(T(h),\mathcal K_+)$ endowed with the topology of $C_{\rm loc}(\mathbb{R},\Phi)$ and the bornology $\mathbb B_{\rm tr}$ described above possesses an attractor $\mathcal A_{\rm tr}\subset\mathcal K_+$ which is strictly invariant and consists of all complete bounded trajectories of (4.1), that is, (4.6) holds. Proof. In contrast to the case with uniqueness, we are now unable to get the result by lifting the global attractor $\mathcal A$ to the trajectory phase space (the dynamical system $(S(t),\Phi)$ does not exist any longer), so we will use Proposition 3.6 instead. Indeed, the shift maps $T(h)$ are continuous in the topology of $C_{\rm loc}(\mathbb{R},\Phi)$ and the dissipative estimate (4.3) gives us a bounded absorbing set $\mathcal B$ for the semigroup $T(h)$. Namely, we may take
$$
\begin{equation*}
\mathcal B:=\biggl\{u\in\mathcal K_+\colon \|u\|_{C_b(\mathbb{R},\Phi)}\leqslant \frac{2C}\alpha\biggr\}.
\end{equation*}
\notag
$$
We claim that this set is actually compact in $\mathcal K_+\subset C_{\rm loc}(\mathbb{R}_+,\Phi)$. Indeed, from equation (4.1), we conclude that $u'$ is uniformly bounded with respect to $u\in\mathcal B$ and Arzelà’s theorem gives the desired compactness. Thus, we have proved the existence of a trajectory attractor $\mathcal A_{\rm tr}\subset\mathcal B$, which is strictly invariant with respect to $T(h)$ and is generated by all complete bounded trajectories of (4.1) (being pedantic, we have verified only that it is generated by all complete bounded trajectories of $T(h)$, but it is immediate to see that there is a one-to-one correspondence between such trajectories and complete bounded solutions of (4.1)). The proposition is proved. Example 4.3. We now consider (following mainly [227]) a bit more complicated case of a reaction-diffusion equation in a bounded domain $\Omega\subset\mathbb{R}^d$: It is not difficult to verify, using, for example, Galerkin approximations and the energy identity (4.10), that weak solutions thus defined exist for every $u_0\in L^2(\Omega):=\Phi$ and satisfy the dissipative estimate
$$
\begin{equation}
\|u(t)\|^2_{L^2}+\int_t^{t+1}\|\nabla_x u(s)\|^2_{L^2}\,ds+ \int_t^{t+1}\|u(s)\|^{p+1}_{L^{p+1}}\,ds\leqslant C\|u_0\|^2_{L^2}\,e^{-\beta t}+C_*
\end{equation}
\tag{4.11}
$$
for some positive constants $C$, $C_*$, and $\beta$ (see [38], [227]). Let us now turn to attractors. As before, we define the trajectory phase space $\mathcal K_+\subset\Theta_{\rm loc}$ as the set of all weak solutions of problem (4.7). Then the semigroup of time shifts $T(h)$ acts on $\mathcal K_+$, so the trajectory dynamical system $(T(h),\mathcal K_+)$ is well defined. As in the previous example, we lift the standard bornology of the phase space $\Phi=L^2(\Omega)$ to the set $\mathcal K_+$, namely, $B\in\mathbb B$ if and only if $B\big|_{t=0}$ is bounded in $\Phi$. Then estimate (4.11) and the embedding guarantee that the set
$$
\begin{equation*}
\mathcal B:=\bigl\{u\in\mathcal K_+\colon \|u\|_{L^2_b(\mathbb{R}_+,W^{1,2}_0)}^2+ \|u\|_{L^{p+1}_b(\mathbb{R}_+, L^{p+1})}\leqslant 2C_*\bigr\}
\end{equation*}
\notag
$$
is a bounded absorbing set for the semigroup $T(h)$. Moreover, since this set is bounded in the reflexive locally convex space $\Theta_{\rm loc}$, it is precompact in the space $\Theta_{\rm loc}^w$ (which is the space $\Theta_{\rm loc}$ endowed with the weak topology) by the Banach–Alaoglu theorem. The set $\mathcal B$ is also closed in $\Theta_{\rm loc}^w$ since it is precompact and sequentially closed (the last fact is proved similarly to the proof of the existence of a weak solution; see [38]), so it is compact by the Eberlein–Smulian theorem (actually, we may avoid the use of the Eberlein–Smulian theory here since $\mathcal B$ is bounded in $\Theta_{\rm loc}$ and the weak topology on it is metrizable). Thus, we have found a compact bounded absorbing set $\mathcal B$ for the trajectory dynamical system $(T(h),\mathcal K_+)$ endowed with the weak topology. Note also that the shift semigroup $T(h)$ is obviously continuous. Then, due to Proposition 3.6, there exists a weak trajectory attractor $\mathcal A_{\rm tr}^w\subset \mathcal K_+$, which is the attractor of the trajectory dynamical system $(T(h),\mathcal K_+)$ in the weak topology of $\Theta_{\rm loc}$. It is strictly invariant and is generated by all complete bounded solutions of equation (4.7). We now prove that the trajectory attractor constructed is actually an attractor in the strong topology of $\Theta_{\rm loc}$ as well. To this end, we need an absorbing set which will be compact in the strong topology as well. We claim that $\mathcal B_1:=T(1)\mathcal B$ is such a set. To verify compactness, we use the energy method developed in [14], [227], [181]. This method is based on the fact that for many Banach spaces (including Hilbert ones, the spaces $L^p$ with $1<p<\infty$, and so on) the weak convergence $\xi_n\to\xi$ in combination with the convergence of the norms imply strong convergence. Let $u_n\in\mathcal B$ be a sequence of solutions. Then without loss of generality we may assume that $u_n\to u\in\mathcal K_+$ in the weak topology of $\Theta_{\rm loc}$. To verify the compactness of $T(1)\mathcal B$ in the strong topology, it is sufficient to prove that $u_n\to u$ strongly in $L^2(T,T+1;W^{1,2}_0)$ and in $L^{p+1}(T,T+1;L^{p+1})$ for every $T\geqslant 1$. For simplicity we consider only the case $T=1$ (the general case is analogous). To do this we multiply the energy identity (4.10) for $u_n$ by $t$ and integrate with respect to $t\in[0,2]$ to get that
$$
\begin{equation}
\begin{aligned} \, \notag &\frac{1}{2}\|u_n(2)\|^2_{\Phi}+ \int_0^2t(a\nabla_x u_n(t),\nabla_x u_n(t))\,dt+ \int_0^2t(f(u_n(t)),u_n(t))\,dt \\ &\qquad=\frac{1}{2}\int_0^2\|u_n(t)\|^2_\Phi\,dt. \end{aligned}
\end{equation}
\tag{4.12}
$$
Using the boundedness of $u_n$ in $\Theta_{\rm loc}$ together with the control (4.9) of time derivatives $\partial_t u_n$, we conclude that $u_n\to u$ weakly in $C(0,2;\Phi)$ and strongly in $L^2(0,2;\Phi)$. Thus, without loss of generality we also have the convergence $u_n\to u$ almost everywhere. Passing to the subsequence if necessary, we extract from the last equality that
$$
\begin{equation}
\begin{aligned} \, \notag &\frac{1}{2}\lim_{n\to\infty}\|u_n(2)\|^2_{\Phi}+ \lim_{n\to\infty}\int_0^2t(a\nabla_x u_n(t),\nabla_x u_n(t))\,dt+ \alpha\lim_{n\to\infty}\int_0^2t\|u_n(t)\|^{p+1}_{L^{p+1}}\,dt \\ &\qquad+\lim_{n\to\infty}\int_0^2t(f(u_n(t)),u_n(t))- \alpha t\|u_n(t)\|^{p+1}_{L^{p+1}}\,dt= \frac{1}{2}\int_0^2\|u(t)\|^2_\Phi\,dt. \end{aligned}
\end{equation}
\tag{4.13}
$$
Using now the weak lower semicontinuity of convex functions and Fatou’s lemma (in order to handle the term containing the nonlinearity $f$) in combination with identity (4.12) for the limit solution $u(t)$, we conclude that
$$
\begin{equation*}
\lim_{n\to\infty}\int_0^2t(a\nabla_x u_n(t),\nabla_x u_n(t))\,dt= \int_0^2t(a\nabla_x u(t),\nabla_x u(t))\,dt
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\lim_{n\to\infty}\int_0^2t\|u_n(t)\|^{p+1}_{L^{p+1}}\,dt= \int_0^2t\|u(t)\|^{p+1}_{L^{p+1}}\,dt.
\end{equation*}
\notag
$$
Taking weak convergence into account, we conclude that $u_n\to u$ strongly in $L^2(1,2;W^{1,2}_0)$ as well as in $L^{p+1}(1,2;L^{p+1})$. Thus, we have verified the compactness of the set $\mathcal B_1$ in the strong topology of $\Theta_{\rm loc}$. In turn, this gives the existence of a trajectory attractor $\mathcal A_{\rm tr}$ in the strong topology of $\Theta_{\rm loc}$ which is generated by all complete bounded trajectories of equation (4.7) and coincides with the weak trajectory attractor constructed above. Remark 4.4. We recall that the possible non-uniqueness of solutions of (4.7) is formally caused by the absence of the local Lipschitz continuity of the nonlinearity $f$. However, in contrast to ODEs, adding this natural assumption would not change the situation drastically due to possible blow up of smooth solutions which may occur despite the dissipative energy estimate (4.11); see [21] for the case of the 3D complex Ginzburg–Landau equation, and see [98], [15], [186] for different classes of reaction-diffusion systems. Remark 4.5. We emphasize another important property of system (4.7) which does not hold for more general classes of PDEs, namely, the fact that every weak solution of (4.7) satisfies the energy identity (4.10) and energy estimate (4.11). In particular, this fact allows us to define the bornology in the trajectory phase space $\mathcal K_+$ just by lifting bounded sets in the classical phase space $\Phi$ to the space of trajectories (see (4.5)), which is not typical for the theory of trajectory attractors (see examples below). As a result, we may define the generalized multi-valued semigroup $S(t)\colon\Phi\to 2^{\Phi}$ via 4.2. Trajectory attractors for elliptic PDEsWe continue by a more simple and a bit artificial example, which demonstrates some important features of trajectory attractors and gives us a toy example for trajectory attractors of elliptic PDEs. Example 4.6. Consider the following second-order ODE: Let us define the trajectory phase space $\mathcal K_+\subset \Theta_{\rm loc}:=C^1_{\rm loc}(\mathbb{R})$ as the set of all solutions $u(t)$, $t\geqslant 0$, of problem (4.15) which correspond to all $u_0\in\mathbb{R}$. The bornology $\mathbb B$ on it is defined as follows: $B\subset\mathcal K_+$ is bounded if $B\big|_{[0,1]}$ is bounded in $C^1[0,1]$. Then the dissipative estimate (4.17) can be rewritten in the following form:
$$
\begin{equation*}
\|u\|_{C^1[T,T+1]}\leqslant C\|u\|_{C^1[0,1]}^2e^{-\kappa T}+C_*
\end{equation*}
\notag
$$
and, therefore, the set $\mathcal B:= \{u\in\mathcal K_+\colon\|u\|_{C^1_b(\mathbb{R}_+)}\leqslant 2C_*\}$ is a bounded absorbing set for the trajectory dynamical system $(T(h),\mathcal K_+)$. Moreover, expressing the second derivative $u''$ from the equation, we see that $u''$ is also bounded if $u\in\mathcal B$, so by Arzelà’s theorem $\mathcal B$ is compact in $\Theta_{\rm loc}$. Thus, according to the general theory, the trajectory dynamical system $(T(h),\mathcal K_+)$ possesses an attractor $\mathcal A_{\rm tr}\subset\mathcal K_+$, which is generated by all complete bounded solutions of (4.15). The structure of the constructed attractor can easily be understood since we have a global Lyapunov function (due to the identity (4.16)), so we have only the equilibria $u=0,\pm1$ and heteroclinic orbits between them. Moreover, since the map $u\to(u(0),u'(0))$ is one-to-one as a map from $\mathcal K_+$ to $\mathbb{R}^2$, the attractor obtained is actually two-dimensional and looks qualitatively like in Example 2.6 (Fig. 1, (a), up to a rotation through $\pi/4$). Thus, the trajectory approach allows us to restore the “forgotten” initial condition on $u'\big|_{t=0}$ and end up with the standard attractor for this equation. Remark 4.7. We see that in the previous example we are unable to obtain a bornology on the trajectory phase space just by lifting the bounded sets from the space $\mathbb{R}$ of the initial data (we will not have the dissipative estimate for such “bounded” sets since $u'(0)$ will be out of control) and the concatenation property is also lost in this case. Actually, we may define a topology and a bornology on $\mathcal K_+$ in several equivalent ways. For instance, we may take $\Theta_{\rm loc}:=C_{\rm loc}(\mathbb{R}_+)$ and understand the first and second derivative of $u$ in the distributional sense (it is not difficult to show that such distributional solutions are actually classical ones and a posteriori $u\in C^2_{\rm loc}(\mathbb{R}_+)$). Bounded sets in $\mathcal K_+$ can also be defined alternatively as sets which are bounded in the Fréchet space $\Theta_{\rm loc}$ or in the Banach space $C^1_b(\mathbb{R})$ (indeed, the dissipative estimate (4.17) guarantees that in all these cases we have the same bornology on $\mathcal K_+$). We mention here a slightly exotic construction of $\mathcal K_+$ and the associated bornology proposed by Vishik and Chepyzhov to handle trajectory attractors for damped wave equations and the 2D damped Euler equations (see [36], [38]). Namely, let us define the trajectory space $\mathcal K_+$ as the set of all solutions $u$ of (4.15) which satisfy the following estimate:
$$
\begin{equation*}
\|u\|_{C^1[T,T+1]}\leqslant C_ue^{-\kappa t}+2C_*,\qquad T\in\mathbb{R}_+,
\end{equation*}
\notag
$$
where $C_*$ and $\kappa$ are fixed (the same as in (4.17)) and the constant $C_u$ may depend on $u$. Note that the set $\mathcal K_+$ thus defined is shift invariant (so the trajectory dynamical system $(T(h),\mathcal K_+)$ is well posed), but a priori it may be smaller than the set of all solutions of (4.15). However, for our model example, thanks to (4.17) we know that a posteriori this is not true. The bornology on $\mathcal K_+$ is then naturally defined as follows: $B\in\mathbb B$ if and only if $C_u\leqslant C_B<\infty$ for all $u\in B$. It is also easy to see that in our case this choice of $\mathbb B$ gives the same bounded sets as the previous constructions. Example 4.8. We now turn to a bit more interesting example of trajectory attractors related to elliptic boundary problems in cylindrical domains. Namely, consider the following elliptic boundary value problem: where $\Omega$ is a bounded domain of $\mathbb{R}^d$, $u=(u^1(t,x),\dots,u^n(t,x))$ is an unknown vector-valued function, $a=a^*>0$ is a given diffusion matrix, $\gamma\in\mathbb{R}$ is a given parameter, and $f\in C(\mathbb{R}^d,\mathbb{R}^d)$ is a given nonlinearity which satisfies the dissipativity assumption for some positive $C$, $\alpha$, and $\varepsilon$. We emphasize that, originally, the PDE (4.18) is not an evolutionary one, but it may appear, for instance, in the study of travelling wave solutions for evolutionary PDEs. Then the parameter $\gamma$ is naturally interpreted as a wave-speed (see [6], [25], [173], [225], [229], and the references therein). One possible and rather popular approach to study these equations is related to the interpretation of the variable along the axis of the cylinder as time and apply the methods of the theory of dynamical systems (centre manifolds, inertial manifolds, attractors, and so on; see [6], [7], [119], [172] for more details). Since the Cauchy problem is ill posed for elliptic equations, as a rule, we have no uniqueness of solutions for problem (4.18) and, for this reason, it is natural (following [225]) to use the trajectory approach. To this end we fix $\Phi:=C_0(\Omega)$ as the space of initial data and consider the set $\mathcal K_+\subset \Theta_{\rm loc}:=C_{\rm loc}(\mathbb{R}_+,\Phi)$ of all (weak) solutions $u(t,x)$ which are defined in the whole of the semi-cylinder $\mathbb{R}_+\times\Omega$. Note from the very beginning that due to the interior regularity for elliptic equations, any solution $u\in W^{2,p}((t,t+1)\times\Omega)$ for all $p<\infty$ and the regularity of a solution is restricted by the smoothness of $f$ only. The following lemma gives the crucial dissipative estimate for solutions of (4.18). Lemma 4.9. Let $u$ be a solution of problem (4.18) defined on the interval $t\in[0,N]$. Then the following estimate holds: where $C_*$ and $Q$ are a constant and a monotone increasing function, respectively, which are independent of $u$, $t$, and $N$, and $H(z)$ is the standard Heaviside function. Sketch of the proof. Let $w(t,x):=au(t,x).u(t,x)$. Then, taking the dot product of equation (4.18) with $u(t,x)$, we arrive at
$$
\begin{equation*}
\begin{aligned} \, \frac{1}{2}(\partial^2_t w(t,x)+\Delta_x w(t,x))&=f(u(t,x)).u(t,x)+ a\nabla_{t,x}u(t,x).\nabla_{t,x}u(t,x) \\ &\qquad-\gamma\partial_t u(t,x).u(t,x) \end{aligned}
\end{equation*}
\notag
$$
and using the dissipativity condition on $f$ and the positivity of the matrix $a$ we get
$$
\begin{equation*}
\partial_t^2 w(t,x)+\Delta_x w(t,x)+ \alpha w(t,x)^{1+\varepsilon/2}\geqslant -C.
\end{equation*}
\notag
$$
Thus, due to the maximum/comparison principle it is enough to verify an analogue of estimate (4.20) for the following ODE: But such an estimate for the ODE is straightforward and we leave it to the reader, so the lemma is proved. The estimate proved allows us to verify the existence of a solution for problem (4.18) for every $u_0\in\Phi$. Indeed, to this end we first solve the corresponding boundary value problem on the finite interval $t\in[0,N]$ with an extra boundary condition $u_N\big|_{t=N}=0$ and then pass to the limit $N\to\infty$. The possibility to do this is guaranteed by this estimate (see [225] for more details). Moreover, passing to the limit $N\to\infty$ in (4.18), we get that any solution $u\in\mathcal K_+$ satisfies the estimate
$$
\begin{equation}
\|u(t)\|_{\Phi}\leqslant Q(\|u_0\|_\Phi) H(1-t)+C_*.
\end{equation}
\tag{4.21}
$$
In particular, any solution $u\in\mathcal K_+$ is bounded as $t\to\infty$. Thus, the trajectory dynamical system $(T(h),\mathcal K_+)$ associated with equation (4.18) is constructed. We endow the space $\mathcal K_+$ with the topology of $\Theta_{\rm loc}$ and with the bornology $\mathbb B$ lifted from the phase space $\Phi$, namely, $B\subset\mathcal K_+$ is in $\mathbb B$ if $B\big|_{t=0}$ is bounded in $\Phi$. Then estimate (4.21) guarantees that the set
$$
\begin{equation*}
\mathcal B:=\{u\in\mathcal K_+\colon \|u\|_{C_b(\mathbb{R}_+,\Phi)}\leqslant C_*\}
\end{equation*}
\notag
$$
is a bounded absorbing set for the trajectory dynamical system defined above. Moreover, due to the interior regularity estimate for elliptic equations, we know that the set $\mathcal B_1:=T(1)\mathcal B$ is bounded in $C^1_b(R_+\times\Omega)$ and, by Arzelà’s theorem, it is compact in $\Theta_{\rm loc}$. Thus, a compact bounded absorbing set for the trajectory dynamical system is constructed and the following result holds. Proposition 4.10. Under the above assumptions problem (4.18) possesses a trajectory attractor $\mathcal A_{\rm tr}\subset\Theta_{\rm loc}$, which consists of all bounded solutions of (4.18) defined on the whole of the cylinder $\mathbb{R}\times\Omega$: Remark 4.11. Note that, at least for application to travelling waves, exactly the set $\mathcal K$ of all bounded solutions of (4.18) on the whole cylinder consists of travelling waves and is the main object of interest. This set clearly contains the equilibria $\mathcal R$ of (4.18) (trivial solutions which are independent of $t$) and it is an interesting and important question whether or not it contains anything else (whether or not non-trivial travelling waves exist). In the case of evolutionary equations we know that the attractor is usually connected (see also [221], [123] and the material below for the connectedness of attractors without uniqueness), and therefore we would expect that $\mathcal K\ne\mathcal R$ if $\mathcal R$ is disconnected. Surprisingly, this may be not true for the attractors of elliptic equations. Indeed, consider equation (4.18) with Neumann boundary conditions, $\gamma=0$, and $f(u)=u(u-1)^2\cdots(u-N)^2$. Then, introducing the function $z(t):=\displaystyle\int_\Omega a u(t,x).u(t,x)\,dx$, we get So the function $z(t)$ is bounded and convex on $\mathbb{R}$, and therefore it is a constant. Since $z''(t)\equiv0$, we conclude that $\nabla_{t,x}u\equiv0$ and $u=\operatorname{const}$. Thus, $\mathcal A_{\rm tr}=\mathcal R=\{0,1,2,\dots,N\}$ is totally disconnected, and we do not have non-trivial travelling waves. Fortunately, the above example is an exception, rather than a rule. Indeed, as proved in [74], we have $\mathcal K\ne\mathcal R$ under some extra mild assumptions, namely, we need to have at least two non-degenerate (hyperbolic) equilibria to guarantee the existence of non-trivial travelling waves. Note that in the above counterexample we have exactly one non-degenerate equilibrium $u=0$ and all others are degenerate. The proof of this fact is based on the analogous result for the Galerkin system of ODEs approximating (4.18) (where it can be established using the Conley index) and the upper semicontinuity of the corresponding trajectory attractors. Remark 4.12. We note that in the case of equation (4.18) the corresponding bornology on the trajectory phase space is defined via lifting the bornology on the usual phase space $\Phi$, exactly as in the case of the reaction-diffusion system considered above. Nevertheless, we do not have the concatenation property here and cannot define the generalized multi-valued semigroup $S(t)$ analogously to (4.14). The reason is that we cannot guarantee that the solution $u(t)$ of (4.18) which is originally defined on the interval $t\in[0,N]$ can be continued to $t\geqslant N$ (it may blow up immediately for $t\geqslant N$), so to define this semigroup properly, we need to use only the solutions which are defined for all $t\geqslant 0$ (interpreting this assumption as some kind of the second boundary condition at $t=\infty$); see [6]. Alternatively, it can be proved that the projection We also mention that the situation becomes much better in the case of so-called fast travelling waves $\gamma\gg1$, where the uniqueness of a solution can be established for problem (4.18). Then the theory becomes very similar to the standard situation related, for example, to reaction-diffusion systems satisfying the assumptions of the uniqueness theorem (see [25], [226]). Remark 4.13. The trajectory approach is also applicable to elliptic boundary value problems in non-cylindrical domains. Indeed, if an unbounded domain $\Omega\subset\mathbb{R}^d$ is invariant with respect to shifts in some fixed direction $\vec l\in\mathbb{R}^d$, that is, 4.3. 3D Navier–Stokes systemIn this subsection we consider various approaches to attractors of the 3D Navier–Stokes system:
$$
\begin{equation}
\partial_t u+(u,\nabla_x)u+\nabla_x p=\nu\Delta_x u+g,\quad \operatorname{div} u=0,\quad u\big|_{t=0}=u_0,
\end{equation}
\tag{4.22}
$$
in a bounded domain $\Omega\subset\mathbb{R}^3$ with smooth boundary. Here $u=(u^1,u^2,u^3)$ is an unknown velocity vector field, $p$ is an unknown pressure, $\nu>0$ is a given viscosity, and $g$ is given external forces. As usual, we denote by $\mathcal V:=\{\varphi\in C_0^\infty(\Omega)\colon \operatorname{div}\varphi=0\}$ the space of divergence-free test functions. Then the phase space $\Phi=H$ and the space $V$ are defined as the closure of $\mathcal V$ in $[L^2(\Omega)]^3$ and $[H^1(\Omega)]^3$, respectively, and the space $V^{-1}$ is the dual space of $V$ with respect to the duality generated by the standard inner product of $H$. We also assume that $u_0\in H$ and $g\in V^{-1}$ (see, for example, [12], [215] for more details). By definition, a function $u\in L^\infty(\mathbb{R}_+,H)\cap L^2_b(\mathbb{R}_+,V)$ is a weak energy solution of (4.22) if, for every test function $\phi\in C_0^\infty(\mathbb{R}_+\times\Omega)$ such that $\operatorname{div} \phi=0$, the following identity holds:
$$
\begin{equation*}
\begin{aligned} \, &-\int_\mathbb{R}(u(t),\partial_t\phi(t))\,dt+ \int_\mathbb{R}\bigl((u(t),\nabla_x)u(t),\phi(t)\bigr)\,dt \\ &\qquad+\nu\int_\mathbb{R}(\nabla_x u(t),\nabla_x \phi(t))\,dt= \int_\mathbb{R}(g,u(t))\,dt. \end{aligned}
\end{equation*}
\notag
$$
It is well known that any such solution $u$ belongs to $C([0,T],H_w)$ for all $T\geqslant 0$ and, moreover, $\partial_t u\in L^{4/3}_b(\mathbb{R}_+,L^{4/3}(\Omega))$. (Here $H_w$ denotes the space $H$ endowed with the weak topology.) In particular, the initial data $u\big|_{t=0}=u_0$ are well posed. It is also well known that, for any $u_0\in H$ and $g\in V^{-1}$, equation (4.22) possesses at least one weak energy solution $u$ which is, in addition, continuous at $t=0$ as a function taking values in $H$ (with the strong topology) and satisfies the following energy inequality:
$$
\begin{equation}
\frac{1}{2}\,\frac{d}{dt}\|u(t)\|^2_{H}+ \nu\|\nabla_x u(t)\|^2_{L^2}\leqslant(g,u(t))
\end{equation}
\tag{4.23}
$$
which is understood in the sense of distributions. By definition, an energy solution $u$ which has these extra properties is called a Leray–Hopf (LH) solution of (4.22) (see [104], [152]). In turn, the energy inequality (4.23) is equivalent to the inequality
$$
\begin{equation}
\|u(t)\|^2_{H}+2\nu\int_s^t\|\nabla_x u(\tau)\|^2_{L^2}\,d\tau\leqslant \|u(s)\|^2_{H}+2\int_s^t(g,u(\tau))\,d\tau,
\end{equation}
\tag{4.24}
$$
which holds for almost all $s\in\mathbb{R}_+$ and all $t\geqslant s$. The solution $u(t)$ satisfying this inequality is strongly continuous at $t=0$ (that is, it is a Leray–Hopf solution) if and only if the inequality holds for $s=0$ (that is, $s=0$ is not in the exceptional set of measure zero); see [38], [194], [214] for more details. Note also that, applying Gronwall’s inequality to (4.24), we get the dissipative energy estimate of the form
$$
\begin{equation}
\|u(t)\|_{H}^2+\nu\int_0^{t}e^{-\beta(t-\tau)} \|\nabla_x u(\tau)\|^2_{L^2}\,d\tau \leqslant \bigl(\|u(s)\|^2_{H}-C\|g\|^2_{V^{-1}}\bigr)e^{-\beta(t-s)}+ C\|g\|^2_{V^{-1}}
\end{equation}
\tag{4.25}
$$
which holds for almost all $s\geqslant 0$ and all $t\geqslant s$ with some positive constants $C$ and $\beta$ which are independent of $t$, $s$, $g$, and $u$. We emphasize that the class of Leray–Hopf solutions is a priori not invariant with respect to time shifts. Indeed, if $u$ is a Leray–Hopf solution and $T(s)u$ is a time shift of it, then we cannot guarantee that (4.24) holds for $s$, and therefore $T(s)u$ may be discontinuous at $t=0$. Important for us is the fact that $T(s)u$ is a Leray–Hopf solution for almost all $s\in\mathbb{R}_+$. We also introduce the class of generalized Leray–Hopf solutions as weak solutions which satisfy (4.23), but which are not necessarily strongly continuous at $t=0$. Obviously, such solutions will be shift-invariant and $T(s)u$ is a Leray–Hopf solution for almost every $s\in\mathbb{R}_+$. The facts mentioned above about the solutions of (4.22) give us a base for attractor theory, but the construction of the corresponding trajectory attractor can be realized in several alternative ways. We start with the most general scheme proposed by Vishik and Chepyzhov; see [36] and [38] where it was applied to weakly damped wave equations with fast growing nonlinearities as well as to the 2D damped Euler equations. Example 4.14. Consider the set $\mathcal K_+^{\rm VC}$ of all weak energy solutions $u$ of (4.22) which correspond to all $u_0\in H$ and satisfy the following estimate: Then, due to assumption (4.26), the set
$$
\begin{equation}
\mathcal B:=\biggl\{u\in\mathcal K_+^{\rm VC}\colon \|u(t)\|_{H}^2+\nu\int^{t}_0e^{-\beta(t-s)} \|\nabla_x u(s)\|_{L^2}^2\,ds\leqslant 2C\|g\|^2_{V^{-1}}\biggr\}
\end{equation}
\tag{4.27}
$$
is a bounded absorbing set for the trajectory dynamical system constructed. Moreover, since
$$
\begin{equation*}
L^\infty(0,T;H)\cap L^2(0,T;V)=[L^1(0,T;H)+L^2(0,T;V^{-1})]^*
\end{equation*}
\notag
$$
for all $T$, the set $\mathcal B$ is precompact in the space $\mathcal K_+^{\rm VC}$ endowed with the topology of $\Theta_{\rm loc}^{w^*}$. Since the weak-star topology on $\mathcal B\subset\Theta_{\rm loc}$ is metrizable, to verify that $\mathcal B$ is closed, it is sufficient to verify its sequential closedness. In turn, this can be done by passing to the limit in (4.22), similarly to the proof of the existence of a weak solution (see [38] for more details). Thus, we have verified that $\mathcal B$ is a bounded compact absorbing set for the trajectory dynamical system $(T(h),\mathcal K_+^{\rm VC})$ and, according to the general theory, there exists a trajectory attractor $\mathcal A^{\rm VC}_{\rm tr}\subset \mathcal K_+^{\rm VC}$ which is generated by all complete bounded weak solutions of (4.22), that is, by all weak solutions which are defined for all $t\in\mathbb{R}$ and satisfy the inequality We denote the set of all such solutions by $\mathcal K^{\rm VC}$.Remark 4.15. The attractor $\mathcal A^{\rm VC}_{\rm tr}$ constructed above attracts, in particular, all Leray–Hopf and generalized Leray–Hopf solutions, however, it is somehow “too big” and contains a lot of non-physical solutions. In particular, it depends on the choice of the constant $C$ in (4.26). Indeed, from [20] we know that, given an arbitrary sufficiently regular function $E(t)$, there exists a weak energy solution $u(t)$ of problem (4.22) which satisfies the additional assumption $\|u(t)\|_{H}=E(t)$, $t>0$. Based on this, we see that even the set $\mathcal K^{\rm VC}$ of complete bounded trajectories depends on the choice of the constant $C$. For this reason it is interesting to discuss alternative constructions which allow us to discard most non-physical solutions. Example 4.16. Consider the set $\mathcal K_+^{\rm gLH}$ of all generalized Leray–Hopf solutions. As we have already mentioned, this set is semi-invariant with respect to time shifts, and therefore the corresponding trajectory dynamical system $(T(h),\mathcal K_+^{\rm gLH})$ is well defined. The topology on $\mathcal K_+^{\rm gLH}$ is naturally defined by the embedding in $\Theta_{\rm loc}^{w^*}$ as in the previous case and we only need a bornology. We say that $B\subset\mathcal K_+^{\rm gLH}$ is bounded if $B\big|_{t\in[0,1]}$ is a bounded set in $L^\infty(0,1;H)$. Then, due to estimate (4.25), we have
$$
\begin{equation}
\|u(t)\|_{H}^2+\nu\int_s^te^{-\beta(t-\tau)} \|\nabla_x u(\tau)\|^2_{L^2}\,d\tau \leqslant C\|u\|^2_{L^\infty(s,s+1;H)}e^{-\beta(t-s)}+C\|g\|^2_{V^{-1}}
\end{equation}
\tag{4.29}
$$
already for all $t\geqslant s\geqslant 0$, and therefore
$$
\begin{equation}
\mathcal B:=\biggl\{u\in \mathcal K_+^{\rm gLH}\colon \|u(t)\|_{H}^2+\nu\int_0^{t}e^{-\beta(t-\tau)}\| \nabla_x u(\tau)\|^2_{L^2}\,d\tau \leqslant 2C\|g\|^2_{V^{-1}}\biggr\}
\end{equation}
\tag{4.30}
$$
is a bounded absorbing set for the trajectory dynamical system under consideration. This set is precompact due to the Banach–Alaoglu theorem, and its compactness follows from the standard fact that the weak limit of generalized Leray–Hopf solutions is a generalized Leray–Hopf solution. Thus, according to the general theory, the trajectory dynamical system under consideration possesses an attractor $\mathcal A^{\rm gLH}_{\rm tr}$ which is generated by all bounded Leray–Hopf solutions of (4.22) defined for all $t\in\mathbb{R}$. We denote the set of such solutions by $\mathcal K^{\rm LH}$. Remark 4.17. The attractor $\mathcal A^{\rm gLH}_{\rm tr}$ constructed above looks as the most natural trajectory attractor for the 3D Navier–Stokes equations and appears in slightly different, but equivalent forms in many papers starting from the seminal works of Vishik and Chepyzhov [36] and Sell [208]. For instance, one can replace the topology of $\Theta_{\rm loc}^{w^*}$ by the strong topology of $L^2_{\rm loc}(\mathbb{R}_+,H)$, avoiding the use of weak topologies in definitions. Indeed, as is not difficult to show, using the control of an appropriate norm of the time derivative $\partial_t u$ in equation (4.22) (see, for example, [208]), the topologies of $L^2_{\rm loc}(\mathbb{R}_+,H)$ and $\Theta_{\rm loc}^{w^*}$ coincide on the absorbing ball $\mathcal B$. Moreover, due to the trick with an alternative generalization of Leray–Hopf solutions proposed in [208], one can use bounded sets of $L^2_{\rm loc}(\mathbb{R}_+,H)$ to define a bornology on the trajectory phase space as well. However, we cannot use the bornology generated by the sets of initial data bounded in $H$. As we have already mentioned, estimate (4.25) may fail at $s=0$ for generalized Leray–Hopf solutions, and the boundedness of the set of $u(0)$ does not imply that the corresponding set of trajectories is bounded in $\Theta_{\rm loc}$. Example 4.18. There is an alternative way, which allows us to construct a trajectory attractor using the bornology related to bounded sets in $H$, namely, one may consider the set $\mathcal K_+^{\rm LH}$ of standard (not generalized) Leray–Hopf solutions as a “trajectory phase space”. The problem here is that, as discussed before, $T(h)\mathcal K_+^{\rm LH}$ may be not a subset of $\mathcal K_+^{\rm LH}$, so we need to deal with non-invariant sets of trajectories. The extension of the attractor theory to this case has been developed in [248]. We will not discuss this theory in more details since there is a simple trick which allows us to reduce it to the general scheme considered in the previous section. Namely, consider the whole of the space $\Theta_{\rm loc}^{w^*}$ as the trajectory phase space for problem (4.22). Then it is obvious that the semigroup $T(h)$ of time shifts acts continuously on $\Theta_{\rm loc}^{w^*}$, so the trajectory dynamical system $(T(h),\Theta_{\rm loc}^{w^*})$ is well defined. Of course, at the moment this trajectory dynamical system looks as an abstract nonsense since it is completely unrelated to the initial Navier–Stokes equation. But we still have not introduced a bornology for this dynamical system and exactly the bornology will relate it with the original problem. Namely, a set $B\subset \Theta_{\rm loc}^{w^*}$ belongs to the bornology $\mathbb B$ if 1) $B\subset \mathcal K_+^{\rm LH}$, that is, $B$ consists of the Leray–Hopf solutions of the Navier–Stokes problem; 2) the set $B\big|_{t=0}:=\{u(0),\ u\in B\}$ is bounded in $H$. Indeed, due to the dissipative estimate (4.25) and our choice of a bornology, we see that the compact set (4.30) constructed above is an absorbing set for this new trajectory dynamical system and, therefore, according to the general theory, we have an attractor $\mathcal A_{\rm tr}^{\rm LH}\subset \mathcal B$ (and therefore $\mathcal A_{\rm tr}^{\rm LH}\subset \mathcal K_+^{\rm gLH}$). Since $T(h)$ is continuous, this theory also guarantees that $\mathcal A_{\rm tr}^{\rm LH}$ is strictly invariant and therefore is generated by some subset of complete bounded Leray–Hopf solutions: Note that the general theory does not give us the coincidence of two attractors since we cannot claim that the above absorbing set belongs to $\mathbb B$ (we do not know whether or not $\mathcal B\subset\mathcal K_+^{\rm LH}$). But in our case it follows immediately from the fact that for every $u\in\mathcal K^{\rm LH}$, we have $T(h)u\big|_{t\geqslant 0}\in \mathcal K_+^{\rm LH}$ for almost every $h\in\mathbb{R}$. Thus, and considering the non-invariant spaces of trajectories does not bring anything new to the theory of trajectory attractors for the 3D Navier–Stokes equations. We also mention that this is a lucky exception in the theory of attractors of non-invariant sets of trajectories that we are able to establish the last equality and clarify the structure of the corresponding attractor. In more general situations we usually do not know how this attractor is related to solutions of the equation under consideration and what complete bounded trajectories generate it. This is actually the main drawback of the theory. Example 4.19. We now consider one more approach to trajectory attractors, which was proposed in [241] (see also [39], [90], [179]) for the study of damped wave equations with supercritical nonlinearities and which is based on approximations of the original system by a system for which we have the uniqueness of solutions. We restrict ourselves to the Galerkin approximations of the original 3D Navier–Stokes system. Namely, let $\{e_n\}_{n=1}^\infty$ be the orthonormal (in $H$) system of eigenvectors of the classical Stokes operator $A:=-\Pi\Delta_x$ in $\Omega$ with Dirichlet boundary conditions, and let $P_N$ be the corresponding orthoprojector onto the first $N$ eigenvectors of $A$: Then, given $N\in\mathbb N$, the corresponding Galerkin approximation system reads We now define the trajectory phase space $\mathcal K_+^{\rm gal}\subset\mathcal K_+^{\rm gLH}\subset\Theta_{\rm loc}$ as the set of all generalized Leray–Hopf solutions of (4.22) which can be obtained as weak-star limits of Galerkin solutions:
$$
\begin{equation}
\mathcal K_+^{\rm gal}:=\Bigl\{u\in\Theta_{\rm loc}\colon u=\lim_{k\to\infty}u_{N_k}\Bigr\},
\end{equation}
\tag{4.32}
$$
where the limit is taken in the topology of $\Theta_{\rm loc}^{w^*}$. Since we do not assume that $u_{N_k}(0)\to u(0)$ strongly in $H$, the set $\mathcal K_+^{\rm gal}$ is invariant with respect to time shifts, and therefore the trajectory dynamical system $(T(h),\mathcal K_+^{\rm gal})$ is well defined. We endow this dynamical system with the topology of $\Theta_{\rm loc}^{w^*}$, so, to speak about attractors, it remains to introduce a bornology on $\mathcal K_+^{\rm gal}$. This requires some accuracy since we want the weak-star limit of such solutions, belonging to a bounded set, to be such a solution too, that is, it should be possible to obtain it as a weak-star limit of Galerkin solutions. To this end, following [241], we introduce the so-called $M$-functional:
$$
\begin{equation}
M_u(t):=\inf_{u_{N_k}\rightharpoondown u}\,\liminf_{k\to\infty} \|u_{N_k}(t)\|^2_{H},\qquad u\in\mathcal K_+^{\rm gal}.
\end{equation}
\tag{4.33}
$$
The external infimum is taken over all subsequences of Galerkin solutions which converge to a given $u\in\mathcal K_+^{\rm gal}$. By definition, the $M$-functional is well defined on $\mathcal K_+^{\rm gal}$ and the straightforward arguments show that
$$
\begin{equation}
\begin{gathered} \, {\rm(a)}\quad M_{T(s)u}(t)\leqslant M_u(t+s);\qquad {\rm(b)}\quad \|u(t)\|^2_{H}\leqslant M_u(t); \\ {\rm(c)}\quad M_u(t+s)+\nu\int_s^{t+s}\|\nabla_x u(\tau)\|_{L^2}^2\,d\tau \leqslant CM_u(s)e^{-\beta t}+C\|g\|_{V^{-1}}^2,\quad t,s\geqslant 0. \end{gathered}
\end{equation}
\tag{4.34}
$$
The key property of the $M$-functional is stated in the lemma below. Lemma 4.20. Let the sequence $u_n\in\mathcal K_+^{\rm gal}$ be such that $u_n\to u$ in $\Theta_{\rm loc}^{w^*}$. Then $u\in\mathcal K_+^{\rm gal}$ and The proof of this lemma is based on the diagonal procedure and the fact that the topology of $\Theta_{\rm loc}^{w^*}$ is metrizable on bounded subsets of $\Theta_{\rm loc}$ (see [241] for the details). We are now ready to define a bornology $\mathbb B$ for the trajectory dynamical system $(T(h),\mathcal K_+^{\rm gal})$. Namely, a set $B\subset\mathcal K_+^{\rm gal}$ is an element of $\mathbb B$ if and only if Then, according to Lemma 4.20 and the dissipative estimate (4.34), the set
$$
\begin{equation*}
\mathcal B:=\{u\in\mathcal K_+^{\rm gal}\colon M_u(0)\leqslant 2C\|g\|^2_{V^{-1}}\}
\end{equation*}
\notag
$$
is a compact and bounded absorbing set for the trajectory dynamical system $(T(h),\mathcal K_+^{\rm gal})$, so according to the general theory, we have the following result. Theorem 4.21. Under the above assumptions, the trajectory dynamical system $(T(h),\mathcal K_+^{\rm gal})$ associated with the Navier–Stokes problem (4.22) possesses an attractor $\mathcal A_{\rm tr}^{\rm gal}\subset \mathcal A_{\rm tr}^{\rm gLH}$ which is generated by all complete bounded trajectories of (4.22) which can be obtained as a weak-star limit of the corresponding Galerkin approximations. Namely, Remark 4.22. The key advantage of the approach related to approximations and the $M$-functional is that we can use not just the energy estimates, but also other types of estimates available on the level of approximations in order to study the corresponding trajectory attractor. Indeed, this approach was originally suggested in [241] for damped wave equations with supercritical nonlinearities, in order to verify that any complete bounded trajectory $u(t)$, $t\in\mathbb{R}$, of such an equation is actually smooth for $t\leqslant T_u$. This fact follows from the smoothness of the corresponding set of equilibria and the gradient structure of the equation, but its proof requires rather delicate estimates, which can be justified on the level of Galerkin approximations only, so the analogous result is not known for other types of trajectory attractors. As a drawback, we mention that the equality $\mathcal A_{\rm tr}^{\rm gLH}=\mathcal A_{\rm tr}^{\rm gal}$ is not known and the attractor obtained may a priori depend on the way of approximation (for example, on the choice of the Galerkin base). 4.4. Connectedness of trajectory attractorsIt is well known that global attractors of evolutionary PDEs are usually connected (see [12], [215]). This is based on the simple topological fact that an $\omega$-limit set $\omega(B)$ of a connected set $B$ is connected if the corresponding dynamical system is continuous (this may not be true under the closed graph assumption alone; see [185]). However, the situation with trajectory attractors is more delicate since it is a priori not clear whether or not the corresponding trajectory phase space is connected (and as we have seen above, it may be not connected at least for the trajectory attractors of elliptic PDEs). Nevertheless, in many cases the trajectory attractors related to evolutionary PDEs remain connected. We briefly discuss below this theory using the model example of the 3D Navier–Stokes equations (an utilizing ideas from [122], [222]), although the method has a general nature and works in many other cases. The key technical tool here is the following simple lemma. Lemma 4.23. Let $\Phi$ be a metric space, and let $\mathcal B$ be a compact set in it. Assume that a family of connected sets $\gamma_\alpha\subset\mathcal B$, $\alpha\in\mathfrak A$, where $\mathfrak A$ is a topological space, is fixed such that $\operatorname{dist}_\Phi(\gamma_\alpha,\mathcal B)\to0$ as $\alpha\to\alpha_0$. Finally, let $u_1,u_2\in\mathcal B$, and let there exist sequences $\alpha_n\to\alpha_0$ and $u_i^n\in \gamma_{\alpha_n}$, $i=1,2$, such that $u_i^n\to u_i$ as $n\to\infty$, $i=1,2$. Then the points $u_1$ and $u_2$ belong to the same connected component of $\mathcal B$. Here and below $\operatorname{dist}_\Phi(A,B)$ denotes the non-symmetric Hausdorff semi-distance between the sets $A$ and $B$ in $\Phi$. Proof. Although the proof of the lemma is standard, for the convenience of the reader we present it here. Indeed, assume that the statement is wrong, so $u_i\in\mathcal B_i$ where the $\mathcal B_i\subset \mathcal B$ are closed, disjoint, and $\mathcal B=\mathcal B_1\cup\mathcal B_2$. Since $\mathcal B$ is compact, there are disjoint $\varepsilon$-neighbourhoods $\mathcal O_\varepsilon(\mathcal B_i)$ of them for some positive $\varepsilon$. Then, since $\gamma_\alpha\to\mathcal B$ in the sense of upper semicontinuity, we have $\gamma_{\alpha_n} \subset\mathcal O_\varepsilon(\mathcal B_1)\cup \mathcal O_\varepsilon(\mathcal B_2)$ for sufficiently large $n$s. On the other hand, since the $u_i$ are limit points, we must have $\gamma_{\alpha_n}\cap\mathcal O_\varepsilon(\mathcal B_i)\ne\varnothing$, $i=1,2$, if $n$ is large enough. This contradicts the connectedness of $\gamma_{\alpha_n}$ and proves the lemma. This lemma will be used in the situation where all the $\gamma_\alpha$ are continuous curves in $\Phi$: $\gamma_\alpha\colon[a,b]\to\Phi$ and $\lim_{\alpha\to\alpha_0}\gamma_{\alpha,a}=u_1$, $\lim_{\alpha\to\alpha_0}\gamma_{\alpha,b}=u_2$ for some $u_1,u_2\in\mathcal B$. If we succeed in finding such families of curves for any $u_1,u_2\in\mathcal B$, this will imply the desired connectedness of $\mathcal B$. It is important to emphasize that, although $u_1,u_2\in\mathcal B$, the curves $\gamma_\alpha$ need not belong to the set $\mathcal B$. In particular, in what follows the set $\mathcal B$ will consist of trajectories of the dynamical system under consideration and $\gamma_{\alpha,s}\in\Phi$ will be generated by some cleverly chosen approximations for these trajectories. We start with the simplest case of the trajectory dynamical system $(T(h),\mathcal K_+^{\rm gal})$. Proposition 4.24. Let $(T(h),\mathcal K_+^{\rm gal})$ be the trajectory dynamical system associated with the Galerkin solutions of the Navier–Stokes equations that were constructed above. Then the set $\mathcal K_+^{\rm gal}$, the absorbing set $\mathcal B$, and the corresponding trajectory attractor $\mathcal A_{\rm tr}^{\rm gal}$ are connected in the topology of $\Theta_{\rm loc}^{w^*}$. Proof. Fix a stationary point $\overline u$ of the Navier–Stokes problem under consideration. It is well known that $\overline u$ exists and is regular enough ($\overline u\in H^1_0(\Omega)$), so $P_N\overline u\to \overline u$ as $N\to\infty$ strongly in $H^1_0(\Omega)$ and, in particular, in $\Theta_{\rm loc}^{w^*}$. This stationary solution can be obtained as a limit of the Galerkin approximations (4.31) starting with $u_N^0:=P_N\overline u$. This fact can easily be proved using weak-strong uniqueness arguments. Thus, $\overline u\in\mathcal B$.
Now let $u\in\mathcal B$ be another Galerkin solution which belongs to the absorbing set $\mathcal B$, and let us prove that $u$ and $\overline u$ belong to the same connected component of $\mathcal B$. By transitivity this will be enough to establish the connectedness of $\mathcal B$ and finish the proof of the proposition. By the definition of $u$ there exists a sequence of Galerkin approximations $u_{N_k}$ such that $u_{N_k}\to u$ in $\Phi:=\Theta_{\rm loc}^{w^*}$ as $k\to\infty$. Let us construct a continuous curve $\gamma_{k,s}\subset\Theta_{\rm loc}^{w^*}$, $s\in[0,1]$, by fixing $s u_{N_k}(0)+(1-s)P_{N_k}\overline u$ as the initial data for the Galerkin system (4.31) and taking $\gamma_{k,s}(\,\cdot\,)\in\Phi$ as a unique solution of this problem. Then the continuity of this curve with respect to $s$ follows from the continuity of the system of ODEs (4.31) with respect to the initial data and, by construction, $\lim_{k\to\infty}\gamma_{k,1}=u$ and $\lim_{k\to\infty}\gamma_{k,0}=\overline u$. Here we essentially use that the whole sequence $\overline u_N$ of Galerkin approximations to $\overline u$ (not only up to a subsequence) is convergent, and this is the main reason why we take one of the points in $\mathcal B$ as a stationary solution. Thus, to apply the previous lemma, we only need to verify that $\operatorname{dist}_\Phi(\gamma_{k,s},\mathcal B)\to0$ as $k\to\infty$. To show this we observe that by the Banach–Alaoglu theorem, $\bigcup\limits_{k\in\mathbb N, s\in[0,1]}\gamma_{k,s}$ is a precompact set in $\Theta_{\rm loc}^{w^*}$ and, by the definition of $\mathcal B$ any limit point of this set as $k\to\infty$ belongs to $\mathcal B$. This gives the desired upper semicontinuity and allows us to apply the lemma to verify that $\mathcal B$ is connected and finish the proof of the proposition. We now turn to other types of trajectory attractors. Proposition 4.25. The trajectory attractors $\mathcal A_{\rm tr}^{\rm gLH}$ and $\mathcal A_{\rm tr}^{\rm VC}$ are connected in $\Theta_{\rm loc}^{w^*}$. Sketch of the proof. The idea of the proof is similar: construct a family of continuous curves $\gamma_{k,s}$ connecting different points of $\mathcal K_+$ in $\Theta_{\rm loc}^{w^*}$, pass to the limit $k\to\infty$, and use Lemma 4.23. However, we now do not have a canonical way to approximate trajectories of our equation, so we need to proceed in a more accurate way. The key method for constructing such curves was proposed in [122], [222] on the level of multi-valued semigroups, so here we only indicate the straightforward changes which should be made in order to adapt this method to the case of trajectory attractors, leaving the details to the reader.
Note that the Galerkin approximations are no longer appropriate for our purposes, so we consider an alternative, the Leray-$\alpha$ approximations:
$$
\begin{equation}
\begin{gathered} \, \partial_t u_\alpha+(v_\alpha,\nabla_x)u_\alpha+\nabla_x p= \nu\Delta_x u_\alpha+g, \\ \operatorname{div} u_\alpha=0, \qquad v_\alpha=(1-\alpha A)^{-1}u_\alpha,\qquad u_\alpha\big|_{t=0}=u_\alpha(0), \end{gathered}
\end{equation}
\tag{4.35}
$$
where $\alpha>0$ is a regularization parameter. It is well known that this problem possesses a unique solution which depends continuously (in both weak and strong topologies) on the initial data $u_\alpha(0)\in H$ for every $\alpha>0$. Moreover, the solution $u_\alpha$ satisfies all the energy estimates stated above uniformly with respect to $\alpha$ and, as $\alpha\to0$, we have convergence in $\Theta_{\rm loc}^{w^*}$ (up to extracting a subsequence) to generalized Leray–Hopf solutions of the limit Navier–Stokes system (4.22). In addition, if $u_\alpha(0)\to u(0)$ strongly in $H$, then the limit solution will be a Leray–Hopf solution (see [34], [194] for more details).
Let us start with the trajectory dynamical system $(T(h),\mathcal K_+^{\rm gLH})$. It is enough to prove that the absorbing set $\mathcal B$ defined by (4.30) is connected in $\Theta_{\rm loc}^{w^*}$. In turn, in order to verify this fact, it is enough to check that the restriction $\mathcal B_T:=\mathcal B\big|_{t\in[0,T]}$ is connected in $\Theta_T^{w^*}=\Theta_{\rm loc}^{w^*}\big|_{t\in[0,T]}$ for any fixed $T>0$. Now let $u_1,u_2\in\mathcal B$. Given $\alpha>0$, we construct a continuous curve $\gamma_{\alpha,s}\in\Theta_T^{w^*}$, $s\in[-T-1,T]$, as follows. For $s\in[0,T]$ we define
$$
\begin{equation}
\gamma_{\alpha,s}(t):=\begin{cases} u_1(t),& t\leqslant s, \\ u_{\alpha,1}(t),& t\geqslant s, \end{cases}
\end{equation}
\tag{4.36}
$$
where $u_{\alpha,1}(t)$ is a unique solution of the Leray-$\alpha$ approximation (4.35), where $u_{\alpha,1}(s)=u_1(s)$. For $s\in[-1,0]$, $\gamma_{\alpha,s}(t)$ solves (4.35) for the initial data $(s+1)u_1(0)-su_2(0)$. Finally, for $s\in[-T-1,-1]$,
$$
\begin{equation}
\gamma_{\alpha,s}(t):=\begin{cases} u_2(t),& t\leqslant -s-1, \\ u_{\alpha,2}(t),& t\geqslant -s-1, \end{cases}
\end{equation}
\tag{4.37}
$$
where $u_{\alpha,2}(t)$ solves (4.35) for the initial data $u_{\alpha,2}(-s-1)=u_2(-s-1)$. It is not difficult to see that $\gamma_{\alpha,s}$ is indeed a continuous curve in $\Theta_T^{w^*}$ such that $\gamma_{\alpha,T}=u_1$ and $\gamma_{\alpha,-T-1}=u_2$.
As in the previous case, one can show that the set $\bigcup\limits_{\alpha,s}\gamma_{\alpha,s}$ is precompact in $\Theta_T^{w^*}$ and all limit points as $\alpha\to0$ are generalized Leray–Hopf solutions of the limiting Navier–Stokes system and belong to $\mathcal B$ (to verify this we use essentially the fact that the compound trajectories (4.36) and (4.37) satisfy the energy equality (4.23) on the whole time interval $t\in[0,T]$). Thus, by Lemma 4.23, $u_1$ and $u_2$ must belong to the same connected component of $\mathcal B_T$ and $\mathcal B_T$ is connected. We now turn to the case of the trajectory dynamical system $(T(h),\mathcal K_+^{\rm VC})$. The proof in this case is analogous: we use exactly the same continuous curves $\gamma_{\alpha,s}$ and the only difference is that we need to check that the compound trajectories (4.36) and (4.37) satisfy the energy estimate involved in the definition (4.27) uniformly with respect to $\alpha$. The last fact is an immediate corollary of (4.25) for $u_\alpha$. Thus, the connectedness of $\mathcal A_{\rm tr}^{\rm VC}$ is verified and the proposition is proved. Remark 4.26. Note that the concatenation property is not known for the trajectory spaces $\mathcal K_+^{\rm gLH}$ and $\mathcal K_+^{\rm gal}$, nevertheless, the corresponding trajectory attractors are connected. On the other hand, as we can see from the proof, the possibility to construct a compound solutions by “gluing together” two different solutions is crucial for this result. For instance, the concatenation property does not hold for two generalized Leray–Hopf solutions, but it holds if the second solution is continuous at the left-hand endpoint, and this is the key point of the proof. In contrast, in general we are unable to extend a solution of an elliptic PDE considered in Example 4.8, which is defined for $t\in[0,T]$, for $t>T$ by solving the appropriate boundary value problem with the “initial” condition at $t=T$, and this allows the trajectory phase space to be disconnected. 5. Attractors for non-autonomous problemsIn this section, we discuss attractor theory for the dynamical processes related to dissipative PDEs which depend explicitly on time. At this moment there are two major approaches to extend the concept of an attractor to non-autonomous equations: the first treats such an attractor as a time-dependent set $\mathcal A(t)$, which leads naturally to pullback attractors (or kernel sections in the terminology of Chepyzhov and Vishik); see [27], [35], [52], [121], and the references therein; the second approach is based on the reduction of a non-autonomous problem to an autonomous one by an appropriate extension of the phase space and leads to uniform attractors which remain independent of time: see [37], [38], [96]. We also mention that there are several intermediate approaches (for example, the so-called forward attractors), which are significantly more difficult to study and less popular. In addition, they often somehow accumulate the drawbacks of both pullback and uniform attractors and, for this reason, are not considered here (see [96], [123] for the details). We start our exposition with pullback attractors. 5.1. Pullback attractors: a general approachThe aim of this subsection is to give a natural extension of our main result, Theorem 3.3, to the non-autonomous case. First we give a general definition of a dynamical process. Definition 5.1. Let $\{\Phi_\tau\}_{\tau\in\mathbb{R}}$ be a family of Hausdorff topological spaces. Then a family of maps $U(t,\tau)\colon\Phi_\tau\to\Phi_t$, $\tau\in\mathbb{R}$, $t\geqslant\tau$, is a dynamical process if As in the autonomous case, in order to speak about attractors, we need to specify what sets are “bounded”. In the non-autonomous case and under the pullback approach, it is natural to consider bounded sets which also depend on time: $t\to B(t)\in\Phi_t$, $t\in\mathbb{R}$. Namely, we specify a bornology $\mathbb B$ as a system of such time dependent sets: $B=\{B(t)\}_{t\in\mathbb{R}}\in\mathbb B$. We impose a unique requirement on elements $B\in\mathbb B$: for any $t\in\mathbb{R}$, $B(t)\ne\varnothing$. We also mention that the bornology $\mathbb B$ is often referred to as a “universe” in the theory of pullback attractors. However, we prefer to keep the name of “bornology” to be consistent with our general theory for the autonomous case. Definition 5.2. Let $U(t,\tau)$ be a dynamical process in Hausdorff topological spaces $\Phi_t$. A time-dependent set $\mathcal B=\{\mathcal B(t)\}_{t\in\mathbb{R}}$ is a pullback absorbing set with respect to the bornology $\mathbb B$ if for every $B\in\mathbb B$ and every $t\in\mathbb{R}$ there exists $T=T(B,t)>0$ such that Analogously, a time-dependent set $\mathcal B$ is an attracting set with respect to the bornology $\mathbb B$ if for every $B\in\mathbb B$, every $t\in\mathbb{R}$, and every neighbourhood $\mathcal O(\mathcal B(t))$ of the set $\mathcal B(t)$ in the topology of $\Phi_t$, there exists $T=T(B,t,\mathcal O)$ such that Remark 5.3. It may look a bit surprising, but exactly the pullback attraction property is a natural generalization of the attraction property to the non-autonomous case (at least if we treat the attractor as a time-dependent set). Roughly speaking, if we fix a present moment of time $t$ and start the evolution from a bounded set $B(t-s)$ sufficiently far back in the past, then its image $U(t,t-s)B(t-s)$ at present will be arbitrarily close to the attracting set $\mathcal B(t)$. In contrast to this, the forward attraction property (that is, if you start from a bounded set $B(t)$ at the present moment of time, then the image $U(t+s,t)B(t)$ is close to $\mathcal B(t+s)$ for $s$ large enough) is much more delicate and not convenient since it may fail for pullback attractors (see Example 5.10 below). Moreover, the author is not aware about any more or less general constructions which give a “forward” attractor with reasonably good properties. Actually, exactly the lack of forward attraction is a key drawback of the theory of pullback attractors. There are two alternative ways to overcome this drawback: one of them is to consider random external forces with some ergodicity (then forward attraction holds in the sense of convergence in measure), and the alternative is to consider non-autonomous exponential attractors where we have uniform in time attraction; see the exposition below. As in the autonomous case, the theory is based on an appropriate generalization of $\omega$-limit sets. Definition 5.4. Let $U(t,\tau)$ be a dynamical process in the Hausdorff topological spaces $\Phi_t$, $t\in\mathbb{R}$, endowed with some bornology $\mathbb B$. Then the pullback $\omega$-limit set of $B\in\mathbb B$ is defined by Definition 5.5. A time-dependent set $\mathcal A=\{\mathcal A(t)\}_{t\in\mathbb{R}}$ is a pullback attractor for a dynamical process $U(t,\tau)$ in Hausdorff topological spaces $\Phi_t$ endowed with some bornology $\mathbb B$ if 1) for any $t\in\mathbb{R}$ the set $\mathcal A(t)$ is compact in $\Phi_t$; 2) $\mathcal A$ is a pullback attracting set; 3) $\mathcal A$ is a minimal (with respect to inclusion) time-dependent set which has properties 1) and 2). The next theorem is an analogue of Theorem 3.3 and is the main result of this subsection. We mention also that, in the case where $\Phi_t$ are Banach spaces and the bornology $\mathbb B$ consists of uniformly bounded sets, this theorem was proved in [50]. Theorem 5.6. Let a dynamical process $U(t,\tau)$, $t\geqslant \tau\in\mathbb{R}$, acting in Hausdorff topological spaces $\{\Phi_t\}_{t\in\mathbb{R}}$ endowed with some bornology $\mathbb B$, possess a compact pullback attracting set $\mathcal B$. Then the dynamical process $U(t,\tau)$ possesses a pullback attractor $\mathcal A(t)$, $t\in\mathbb{R}$. Moreover, for all $t\in\mathbb{R}$, we have $\mathcal A(t)\subset\mathcal B(t)$. Sketch of the proof. As in the autonomous case, the statement follows from the corresponding properties of $\omega$-limit sets. Namely, we need to verify that under the assumptions of the theorem, for every $B\in\mathbb B$, $\omega_B(t)$ is a non-empty compact subset of $\mathcal B(t)$ that pullback attracts the images of $B$ and that is the minimal compact set possessing this property. Then the desired attractor is defined via the standard formula
$$
\begin{equation*}
\mathcal A(t):= \biggl[\,\bigcup_{B\in\mathbb B}\omega_B(t)\biggr]_{\Phi_t},\qquad t\in\mathbb{R}.
\end{equation*}
\notag
$$
The verification of the above properties of pullback $\omega$-limit sets repeats the proof of Theorem 3.3 almost word by word and, for this reason, is left to the reader. Analogously to the autonomous case, a pullback attractor is strictly invariant with respect to the corresponding dynamical process if the maps $U(t,\tau)$ are continuous. Proposition 5.7. Let the assumptions of Theorem 5.6 hold, and let, in addition, $U(t,\tau)\colon\Phi_\tau\to\Phi_t$ be continuous for all fixed $t\geqslant \tau\in\mathbb{R}$. Then the pullback attractor $\mathcal A(t)$ is strictly invariant: The proof of this statement also repeats word by word the proof of Proposition 3.4 and thus is omitted. Remark 5.8. We see that, analogously to the autonomous case, we have the invariance of the pullback attractor if the maps $U(t,\tau)$ are continuous and this property remains true if we replace continuity by the assumption that the graphs of all maps $U(t,\tau)$ are closed. However, in contrast to the autonomous case, we cannot in general replace the minimality assumption in Definition 5.5 by the strict invariance (5.5) even if all maps $U(t,\tau)$ are continuous. Indeed, as simplest examples show (see, for instance, [27], [179] and Example 5.10 below), such a modification of the definition may lead to the lack of uniqueness of a pullback attractor. We now discuss the validity of the representation formula for pullback attractors. We say that a complete trajectory $u(t)$, $t\in\mathbb{R}$, is bounded ($\mathbb B$-bounded) if $\{u(t)\}_{t\in\mathbb{R}}\in\mathbb B$ and denote by $\mathcal K$ the set of all bounded complete trajectories of the dynamical process $U(t,\tau)$ (following Vishik and Chepyzhov, we refer to $\mathcal K$ as the $\mathbb B$-kernel of the dynamical process $U(t,\tau)$). Then, obviously if the pullback attractor exists. The next statement is a non-autonomous analogue of Proposition 3.6.Proposition 5.9. Let the assumptions of Theorem 5.6 hold, and let, in addition, the maps $U(t,\tau)$ be continuous for all fixed $t\geqslant \tau\in\mathbb{R}$ and the compact absorbing set $\mathcal B$ be bounded ($\mathcal B\in\mathbb B$). Assume also that the bornology $\mathbb B$ is stable with respect to inclusions (if $B\in\mathbb B$ and $B_1(t)\subset B(t)$ for all $t\in\mathbb{R}$ and $B_1(t)\ne\varnothing$, then $B_1\in\mathbb B$). Then where $\mathcal K$ is the $\mathbb B$-kernel of the dynamical process $U(t,\tau)$. The proof of this statement is similar to the proof of Proposition 3.6, and we omit it. Example 5.10. Consider the following ODE in $\Phi_t\equiv\mathbb{R}$: Also fix the standard bornology $\mathbb B$ as the bornology of uniformly bounded sets, namely, $B\in\mathbb B$ if and only if $B(t)\ne\varnothing$ and $\sup_{t\in\mathbb{R}}\|B\|<\infty$. Then, as is not difficult to see, the solution operators $U(t,\tau)$ of the ODE under consideration generate a dynamical process in $\mathbb{R}$ which possesses a pullback attractor $\mathcal A(t)=\{0\}$. Note that this attractor does not possess the forward in time attraction property. Indeed, for $t\geqslant 0$, the equilibrium $y=0$ is exponentially unstable and all close trajectories run away from the neighbourhood of zero (and approach the interval $[-1,1]$, which is a uniform attractor in this case). This example demonstrates the key drawback of the theory of pullback attractors, which forces us to identify the repelling point $y=0$ with an “attractor”. Fix two points $a>0$ and $b<0$ and consider the corresponding solutions $a(t)$ and $b(t)$, $t\in\mathbb{R}$, of equation (5.7) satisfying $a(0)=a$ and $b(0)=b$. Then the time-dependent set has properties 1) and 2) in Definition 5.5 together with the strict invariance (5.5). This example shows that the minimality property of pullback attractors cannot be replaced by strict invariance without the risk to lose the uniqueness of a pullback attractor.5.2. Cocycles and random attractorsThe abstract construction of a pullback attractor is somehow “too general” for practical applications, so it looks reasonable to consider its particular cases. One of the most interesting special cases is given by a cocycle, which, in particular, bridges the gap between the theories of pullback and random attractors. Definition 5.11. Let $\Phi$ be a Hausdorff topological space, $\Psi$ be an arbitrary set, and let $T(h)\colon\Psi\to\Psi$, $h\in\mathbb{R}$, be a group of operators acting on $\Psi$. Then the family of maps $\mathcal S_\xi(t)\colon\Phi\to\Phi$, $\xi\in\Psi$, $t\geqslant 0$, is a cocycle over the group $T(h)$ if As is not difficult to check, any cocycle $\mathcal S_\xi(t)$ generates a family of dynamical processes on $\Phi$ depending on a parameter $\xi\in\Psi$ via
$$
\begin{equation}
U_\xi(t,\tau):=S_{T(\tau)\xi}(t-\tau),\qquad t\geqslant \tau\in\mathbb{R},\quad \xi\in\Psi.
\end{equation}
\tag{5.9}
$$
This family satisfies the additional translation identity
$$
\begin{equation}
U_\xi(t+s,\tau+s)=U_{T(s)\xi}(t,\tau),\qquad \xi\in\Psi, \quad t\geqslant \tau\in\mathbb{R},\quad s\in\mathbb{R}.
\end{equation}
\tag{5.10}
$$
Vice versa, any family of dynamical processes $U_\xi(t,\tau)\colon\Phi\to\Phi$ which satisfies the translation identity (5.10) generates a cocycle $\mathcal S_\xi(t):=U_\xi(t,0)$ in $\Phi$. In applications we are usually given a non-autonomous PDE
$$
\begin{equation}
\partial_t u=A(u,\xi(t)),\quad u\big|_{t=\tau}=u_\tau,
\end{equation}
\tag{5.11}
$$
where $A(\,\cdot\,{,}\,\cdot\,)$ is a nonlinear operator which we do not specify here and $\xi=\xi(t)$ accumulates all terms of the equation that depend explicitly on time, $\Psi$ is an appropriate space of time-dependent functions (for example, some shift-invariant subspace of $L^2_{\rm loc}(\mathbb{R},H)$ where $H$ is a Banach space) and $T(h)\colon\Psi\to\Psi$, $h\in\mathbb{R}$, is the group of time shifts: $(T(h)\xi)(t)=\xi(t+h)$. Then, if problem (5.11) is globally well posed for all $u_\tau\in\Phi$ and all $\xi\in\Psi$, then the corresponding solution operators $U_\xi(t,\tau)\colon\Phi\to\Phi$ generate a translation-invariant family of dynamical processes, and therefore also generate a cocycle over $T(h)\colon\Psi\to\Psi$ in the phase space $\Phi$ (see [27], [38], [123] for more details). In the case of cocycles it is natural to define a bornology $\mathbb B$ as a collection of non-empty $\xi$-dependent sets $\{B(\xi)\}_{\xi\in\Psi}\in\mathbb B$. Then absorbing and attracting sets will be also $\xi$-dependent sets. For instance, the set $\{\mathcal B(\xi)\}_{\xi\in\Psi}$ is pullback attracting if for every $B\in\mathbb B$, every $\xi\in\Psi$, and every neighbourhood $\mathcal O(\mathcal B(\xi))$ of the set $\mathcal B(\xi)$ in $\Phi$, there exists $\tau=\tau(\xi,B,\mathcal O)$ such that
$$
\begin{equation*}
\mathcal S_{T(-t)\xi}(t)B(T(-t)\xi)\subset\mathcal O(\mathcal B(\xi)),\qquad t\geqslant \tau.
\end{equation*}
\notag
$$
Analogously, $\{\mathcal A(\xi)\}_{\xi\in\Psi}$ is a pullback attractor for the cocycle $\mathcal S_\xi(t)$ if 1) the set $\mathcal A(\xi)$ is compact for all $\xi\in\Psi$; 2) the set $\{\mathcal A(\xi)\}_{\xi\in\Psi}$ is a pullback attracting set for $\mathcal S_\xi(t)$; 3) it is the minimal (with respect to inclusion) set which has properties 1) and 2). The next theorem gives the analogue of Theorem 5.6. Theorem 5.12. Let $\Phi$ be a Hausdorff topological space, $\Psi$ be a set, and let $\mathcal S_\xi(t)\colon\Phi\to\Phi$ be a cocycle over the group $T(h)\colon\Psi\to\Psi$, $h\in\mathbb{R}$. Assume that this cocycle possesses a compact pullback attracting set $\{\mathcal B(\xi)\}_{\xi\in\Psi}$. Then there exists a pullback attractor $\{\mathcal A(\xi)\}_{\xi\in\Psi}$ which is a compact subset of $\{\mathcal B(\xi)\}_{\xi\in\Psi}$. Sketch of the proof. This statement is a straightforward corollary of Theorem 5.6. Indeed, for every fixed $\xi\in\Psi$ let us consider the corresponding dynamical process $U_\xi(t,\tau)$ in the phase space $\Phi$ endowed with the bornology $\mathbb B_\xi$ which consists of time dependent sets $B_\xi(t):=B(T(t)\xi)$ for all $B\in\mathbb B$. Then the dynamical process $U_\xi(t,\tau)$ satisfies all the assumptions of Theorem 5.6, and therefore there exists a pullback attractor $\mathcal A_\xi(t)$, $t\in\mathbb{R}$, for this process. Moreover, due to the translation identity and the uniqueness of a pullback attractor, we have $\mathcal A_{T(h)\xi}(t)=\mathcal A_\xi(t+h)$, $t,h\in\mathbb{R}$ and $\xi\in\Psi$. Thus, the $\xi$-dependent set is well defined and gives the desired pullback attractor for the cocycle $\mathcal S_\xi(t)$. As in the previous case, if, in addition, the maps $\mathcal S_\xi(t)$ are continuous for every fixed $t\geqslant 0$ and $\xi\in\Psi$, then we have the strict invariance of the pullback attractor, which now reads
$$
\begin{equation*}
\mathcal S_\xi(t)\mathcal A(\xi)=\mathcal A(T(t)\xi),\qquad \xi\in\Psi,\quad t\geqslant 0.
\end{equation*}
\notag
$$
Moreover, if, in addition, the compact attracting set $\mathcal B\in\mathbb B$ and the bornology $\mathbb B$ is stable with respect to inclusions, then we also have the representation formula where $\mathcal K_\xi$ is the $\mathbb B_\xi$-kernel of the dynamical process $U_\xi(t,\tau)$. As we have seen in Example 5.10, the forward attraction property may fail even in the case of a relatively simple dependence of the symbols $\xi(\,\cdot\,)\in\Psi$ on time. The situation becomes much better if we assume some recurrence properties of time-dependent external forces. Namely, assume that $\Psi$ possesses an invariant (with respect to $T(h)$) probability measure $\mu$, that is, $(\Psi,\mathcal F,\mu)$, where $\mathcal F$ is a $\sigma$-algebra, is a probability space and the maps $T(h)$ are measure preserving. Then it is also natural to assume that the maps $\xi\to \mathcal S_\xi(t)$ are measurable (or even continuous) and for every $B\in\mathbb B$ the map $\xi\to B(\xi)$ is a measurable set-valued map (the sets satisfying this property are called random sets, and the corresponding cocycle is called a random dynamical system (RDS)). Moreover, in order to work with random sets it is usually assumed that $\Phi$ is a Polish space (that is, it is separable, metrizable, and complete). We also mention that it is natural for random dynamical systems that all properties starting from (5.8) do not hold for all $\xi\in\Psi$, but just for $\mu$-almost all of them. The detailed exposition of the theory of random dynamical systems is out of scope of this survey; we refer the interested reader to [52], [51], [56], [55], [123], and the references therein for more details, and restrict ourselves to mentioning the random analogue of Theorem 5.12 and giving some examples. Theorem 5.13. Let $\mathcal S_\xi(t)\colon\Phi\to\Phi$ be an RDS over the measure-preserving group $T(h)\colon\Psi\to\Psi$ acting on the probability space $\Psi$, and let $\Phi$ be Polish. Assume also that this RDS possesses a compact random attracting set $\mathcal B$ (that is, $\mathcal B(\xi)$ is compact for almost all $\xi\in\Psi$) with respect to some bornology $\mathbb B$ which consists of random sets. Then the RDS $\mathcal S_\xi(t)$ possesses a pullback attractor which is a compact random set in $\Phi$. Moreover, if the maps $\mathcal S_\xi(t)$ are continuous for almost all $\xi$, then the attractor $\mathcal A(\omega)$ is strictly invariant. The proof of this theorem is reduced to Theorem 5.12 under the extra assumption that the bornology $\mathbb B$ is separable, that is, there exists a countable bornology $\mathbb B_0$ such that for every $B\in\mathbb B$ there exists $B_0\in\mathbb B_0$ such that $B\subset B_0$ (see [51]). In the general case, when this property fails (for instance when the point random attractors are considered), the proof is more delicate. In particular, it is not a priori clear why the set $\xi\to\Bigl[\,\bigcup\limits_{B\in\mathbb B}\omega_B(\xi)\Bigr]_{\Phi}$, which contains a non-countable union, should be a random set. Moreover, as shown in [56], this union can be significantly larger than the desired random attractor $\mathcal A(\xi)$. In the example given there, the random pullback attractor is a one-point set for almost all $\xi\in\Psi$ and the above union coincides with the whole space $\Phi$ almost surely. Nevertheless, as shown in [56], the theorem remains true in the general setting as well (and even the measurability of sets in $\mathbb B$ can be removed). We also mention the paper [206], where examples of random attractors of a different type are given (in particular, examples where the random attractor does not attract bounded sets forward in time). Corollary 5.14. Under the assumptions of Theorem 5.13, the random attractor $\mathcal A(\xi)$ possesses the forward attraction property in the following sense: for every $B\in\mathbb B$ and every $\varepsilon>0$, where $\operatorname{dist}_{\Phi}(A,B):=\sup_{y\in A}\inf_{x\in B}d(x,y)$ is a non-symmetric Hausdorff distance in $\Phi$. Proof. Indeed, due to the pullback attraction property, for almost all $\xi\in\Psi$, we have
$$
\begin{equation*}
\lim_{t\to\infty} \operatorname{dist}_\Phi(U_{T(-t)\xi}(-t,0)B(\xi), \mathcal A(\xi))=0.
\end{equation*}
\notag
$$
Therefore, by Lebesgue’s dominated convergence theorem, Since $T(t)$ is measure preserving, the change of the independent variable $\xi\to T(t)\eta$ in combination with the translation identity $U_{T(t)\eta}(0,-t)=U_\eta(t,0)$ yields which gives the desired convergence in measure. The corollary is proved. We illustrate the theory by few model examples; more examples can be found, for instance, in [123], [206]. Example 5.15. Let us return to Example 5.10. It can be written in the form of (5.11) with The dynamical system $(T(h),\Psi)$ carries exactly two ergodic invariant probability measures: $\mu_0=\delta(\xi)$ and $\mu_1=\delta(\xi-1)$. The supports of both of these measures correspond to the autonomous equations with attractors $\mathcal A_{\mu_0}=\{0\}$ and $\mathcal A_{\mu_1}=[-1,1]$ both of which are forward “random” attractors. Remark 5.16. We see that, although the measures constructed in the previous example satisfy formally all of the assumptions of the theory, they are a posteriori useless and do not give any additional information about the dynamical system under consideration. This indicates a general problem of the theory of dynamical systems and, in particular, ergodic theory. Namely, although the invariant measure can be constructed on a general compact space using Prohorov’s theorem, this is not enough to get reasonable information about the system under consideration, and to do this we need some “good” properties of this measure (for instance, to have a so-called “physical measure”), for example, its absolute continuity with respect to the Lebesgue measure if $\Psi$ is a subset of $\mathbb{R}^n$. Unfortunately, such a good measure may not exist and it is not clear how to construct it in a more or less general situation (see [116], [235] and the references therein for more details). Example 5.17. Let us consider the damped nonlinear pendulum equation with sign-changing dissipation: To get an RDS, we assume that there exists a Borel measure $\mu$ on the set $\Psi$ which is invariant with respect to shift operators $T(h)$, $h\in\mathbb{R}$, and is ergodic. We also assume that the following dissipativity assumption holds:
$$
\begin{equation}
\kappa_0:=\int_{\gamma\in\Psi}\int_0^1\gamma(t)\,dt\,\mu(d\gamma)>0.
\end{equation}
\tag{5.15}
$$
In order to specify the appropriate bornology $\mathbb B$ on $\Phi$, we need to recall that the function $t\to f(t)$, $t\in\mathbb{R}$, is called tempered if, for every $\beta>0$, (see [27], [120] for more details). We say that a random set $\{B(\xi)\}_{\xi\in\mathbb{R}}$ is tempered if the function $t\to \|B(T(-t)\xi)\|_{\Phi}$ is tempered for $\mu$-almost all $\xi\in\Phi$. The standard bornology which consists of all tempered sets $\{B(\xi)\}_{\xi\in\Psi}$ will be denoted by $\mathbb B_{\rm temp}$. At the next step we use Birkhoff’s ergodic theorem in order to establish that there exists a shift-invariant subset $\Psi_{\rm erg}\subset\Psi$ of full $\mu$-measure such that
$$
\begin{equation}
\lim_{T\to\infty}\frac1T\int_{-T}^0\gamma(t)\,dt=\kappa_0>0,\qquad \gamma\in\Psi_{\rm erg},
\end{equation}
\tag{5.16}
$$
and the existence of this limit allows us to establish that the function
$$
\begin{equation}
b_\gamma(t):=2C_\kappa\int_{-\infty}^t\exp\biggl\{-\int_{m}^t \biggl(2\,\frac{p+2}{p+4}\gamma(\tau)-\kappa\biggr)\,d\tau\biggr\}\,dm
\end{equation}
\tag{5.17}
$$
is well defined and tempered for all $\gamma\in\Psi_{\rm erg}$ if $\kappa>0$ is small enough (the exponent $\kappa$ is fixed from now on in such a way that this condition is satisfied); see [30] for the details. In turn, this fact, together with (5.14), allows us to verify that the tempered random set
$$
\begin{equation}
\mathcal B(\gamma):=\{\xi_y(0)\in\Phi\colon E_{y}(0)\leqslant b_\gamma(0)\}
\end{equation}
\tag{5.18}
$$
is a compact pullback absorbing set for the RDS $\mathcal S_\gamma(t)\colon\Phi\to\Phi$ associated with equation (5.13); see [30]. Thus, according to the general theory, this RDS possesses a tempered random attractor $\mathcal A(\gamma)\subset\mathcal B(\gamma)$, $\gamma\in\Psi_{\rm erg}$, which is strictly invariant and possesses the standard representation formula:
$$
\begin{equation}
\mathcal A(\gamma)=\mathcal K_\gamma\big|_{t=0},\qquad \gamma\in\Psi_{\rm erg},
\end{equation}
\tag{5.19}
$$
where the tempered kernel $\mathcal K_\gamma$ consists of all complete tempered trajectories of equation (5.13). As a natural example of a set $\Psi$ and measure $\mu$ consider the set of random piecewise constant dissipation rates
$$
\begin{equation*}
\gamma_l(t):=l_n,\qquad t\in[n,n+1),\quad n\in\mathbb Z,
\end{equation*}
\notag
$$
where $l=\{l_n\}_{n\in\mathbb Z}\in \Gamma:=\{a,-b\}^{\mathbb Z}$. In other words, the dissipation rate $\gamma_l(t)$ may take only two values $a$ and $-b$ on the interval $[n,n+1)$, $a,b>0$, which are parameterized using the Bernoulli shift scheme $\Gamma$. We assume that the values $a$ and $-b$ have probabilities $q$ and $1-q$ to appear, so we endow $\Gamma$ with the standard Bernoulli product measure $\mu_q$. It is well known (see, for example, [116]) that discrete Bernoulli shifts $\mathcal T(n)\colon\Gamma\to\Gamma$ preserve this measure and that $\mu_q$ is ergodic with respect to these shifts. Using the obvious fact that
$$
\begin{equation}
T(n)\gamma_l=\gamma_{\mathcal T(n)},\qquad l\in\Gamma, \quad n\in\mathbb Z,
\end{equation}
\tag{5.20}
$$
we may lift the dynamical system $(\mathcal T(n),\Gamma)$ to the shift-invariant set $\Psi\subset L^\infty(\mathbb{R})$. Namely, let Then $\Psi$ is obviously shift invariant and is compact in $L^1_{\rm loc}(\mathbb{R})$. The commutation (5.20) also allows us to lift the Bernoulli measure $\mu_q$ to the appropriate shift-invariant measure $\tilde\mu_q$ on $\Psi$. Finally, condition (5.15) now reads and gives us the sufficient condition for the existence of a random attractor $\mathcal A(\gamma)$ (see [30] for the details). Remark 5.18. The above example demonstrates many typical features of attractor theory for random dynamical systems. First, we see the crucial role of Birkhoff’s ergodic theorem or its probabilistic analogues in constructing random attractors. Second, since the existence of a Birkhoff limit (5.16) is not guaranteed for all $\gamma\in\Psi$, but only for almost all of them, the use of “$\mu$-almost all” in all definitions related to random attractors also looks unavoidable. In the exceptional measure-zero choices of $\gamma$ the corresponding dynamical process is simply not dissipative and does not possess an attractor (for example, it will be so if we take $\gamma(t)=-b<0$ for all $t\in\mathbb{R}$). We also mention that, at least in the example related to Bernoulli shifts, the function $b_\gamma(t)$ is unbounded with probability one as $t\to\pm\infty$. This explains why it is important to develop the pullback attractor theory for non-autonomous equations with unbounded in time coefficients (see also next example). The bornology $\mathbb B_{\rm temp}$ of tempered sets introduced above looks as an appropriate and natural generalization of the bornology $\mathbb B_{\rm bound}$ of uniformly bounded in time sets (which is widely used in the deterministic case; see [12], [215]) to the dynamical process related to random and stochastic dynamical systems; see [120], [123]. Note however, that in many cases, the use of the bornology consisting of deterministic (uniformly bounded) sets is enough to recover a random attractor in the proper way; see [52], [56], and the references therein. Our next example will be related to the simplest nonlinear stochastic ODE. Example 5.19. Consider the following stochastic ODE: where $\eta'(t)$ is a two-sided white noise, and $\varepsilon>0$ is a small parameter. Following the standard procedure we realize the two-sided Wiener process on the space $\Psi:=C_0(\mathbb{R})$ of continuous functions $\eta\colon\mathbb{R}\to\mathbb{R}$ which are equal to zero at $t=0$ endowed with the locally compact topology and the standard Wiener measure $\mu$. Then the modified group of temporal shifts $(T(h)\eta)(t):=\eta(t+h)-\eta(h)$ acts on $\Psi$ and the Wiener measure $\mu$ is invariant and ergodic (see, for example, [212]). We present the solution $y(t)=y_\eta(t)$ in the form of $y_\eta(t)=\varepsilon v_\eta(t)+w_\eta(t)$, where $v_\eta(t)$ is the stationary Ornstein–Uhlenbeck process:
$$
\begin{equation}
v_\eta'+v_\eta=\eta'(t),\quad \text{that is,}\quad v_\eta(t)=\eta(t)-\int_{-\infty}^t\eta(s)e^{s-t}\,ds,
\end{equation}
\tag{5.22}
$$
and the remainder solves
$$
\begin{equation}
w'_\eta-w_\eta+(\varepsilon v_\eta(t)+w_\eta)^3=2\varepsilon v_\eta(t),\qquad w_\eta\big|_{t=0}=y_0-\varepsilon v_\eta(0).
\end{equation}
\tag{5.23}
$$
Thus, the random dynamical system $\mathcal S_\eta(t)\colon\mathbb{R}\to\mathbb{R}$ over the group of shifts $T(h)\colon\Psi\to\Psi$ is well defined and we may speak about its pullback random attractor $\mathcal A(\eta)$. According to the general theory, to this end we only need to construct a compact random absorbing set $\mathcal B(\eta)$ with respect to the bornology $\mathbb B_{\rm temp}$ of tempered random sets on $\Phi=\mathbb{R}$. In turn, this means that we need to construct such an absorbing set for equation (5.23) for almost all $\eta$. This can be done in many ways; for instance, multiplying the equation by $w_\eta(t)$ and using Young’s inequality, one easily derives that
$$
\begin{equation*}
\frac{d}{dt}(w_\eta^2)+w_\eta^2\leqslant C(1+\varepsilon^4v_\eta(t)^4)
\end{equation*}
\notag
$$
for some deterministic constant $C$ which is independent of $\varepsilon$. Therefore, for all $t-s\geqslant 0$ we have
$$
\begin{equation}
w_\eta(t)^2\leqslant w_\eta(s)^2e^{s-t}+C\int_{s}^t e^{\kappa-t}(1+\varepsilon^4v_\eta(\kappa)^4)\,d\kappa.
\end{equation}
\tag{5.24}
$$
It only remains to note that the function
$$
\begin{equation*}
b_\eta(t):=C\int_{-\infty}^t e^{\kappa-t}(1+\varepsilon^4v_\eta(\kappa)^4)\,d\kappa
\end{equation*}
\notag
$$
is tempered for almost all $\eta$, as also is the function $v_\eta(t)$ and, consequently the random set
$$
\begin{equation*}
\mathcal B(\eta):=v_\eta(0)+\bigl\{y_0\in\mathbb{R}\colon y_0^2\leqslant b_\eta(0)\bigr\}
\end{equation*}
\notag
$$
is a compact tempered pullback absorbing set for the RDS $\mathcal S_\eta(t)$ constructed, which is associated with equation (5.22). Thus, according to the general theory, this system possesses a tempered pullback random attractor $\mathcal A_\varepsilon(\eta)$ which is generated by all complete tempered trajectories of (5.22) (see [53] for the details). The RDS constructed possesses some remarkable properties. Namely, solving the associated Kolmogorov–Fokker–Planck equation, we see that
$$
\begin{equation}
\nu(dy)=F_\varepsilon(y)\,dy,\quad F_\varepsilon(y):=\frac{\exp\{\varepsilon^{-1}(-y^4/4+y^2/2)\}} {\int_{\mathbb{R}}\exp\{\varepsilon^{-1}(-y^4/4+y^2/2)\}\,dy}\,,
\end{equation}
\tag{5.25}
$$
is an invariant (stationary) measure for the stochastic process under consideration. Moreover, this measure is uniquely ergodic and mixing (see [53], [117] for the details). Due to the ergodicity, the corresponding Lyapunov exponent is given by
$$
\begin{equation}
\lambda(\varepsilon):=\int_{\mathbb{R}}(1-3y^2)F_\varepsilon(y)\,dy= -\frac{1}{2}-\frac{3}{2}\, \frac{I_{-3/4}(1/(8\varepsilon))+I_{3/4}(1/(8\varepsilon))} {I_{-1/4}(1/(8\varepsilon))+I_{1/4}(1/(8\varepsilon))}\leqslant -\frac{1}{2}<0,
\end{equation}
\tag{5.26}
$$
where $I_\kappa(z)$ is the modified Bessel function of the first kind and order $\kappa$. Since the Lyapunov coefficient is negative for all $\varepsilon>0$, the attractor $\mathcal A_\varepsilon(\eta)$ consists of a single point $\mathcal A_\varepsilon(\eta)=\{u_{\varepsilon,\eta}(0)\}$ and the corresponding tempered kernel $\mathcal K_{\varepsilon,\eta}$ consists of a single exponentially stable trajectory $u_{\varepsilon,\eta}(t)$ for almost all $\eta>0$. Thus, adding an arbitrary small additive noise destroys the pitchfork instability and makes the random system exponentially stable. In particular, the limit deterministic attractor $\mathcal A_0=[-1,1]$ is not robust with respect to random perturbations (see [53] for more details and a related discussion). Remark 5.20. The stabilization effect of adding a small additive noise appears in much more general situations, including stochastic reaction-diffusion equations, and so on. The key ingredients here are: 1) the uniqueness and ergodicity of the invariant measure which is known in more or less general situation if the noise is not too degenerate (see [117], [142], and the references therein); 2) the order-preserving structure which allows us to construct the maximal and minimal solutions belonging to the attractor. If these two solutions coincide almost surely, then the attractor consists of a single point as in the above example, and if not, then every of them carries an invariant measure which contradicts uniqueness (see [42] for the details). Note that this mechanism does not work if the limiting deterministic dynamics is really chaotic and have positive Lyapunov exponents, for instance, in the case of the Lorenz system (2.9) perturbed by a small additive noise, so we expect non-trivial attractors and rich dynamics in the stochastic case as well. Unfortunately, despite the solid numerical evidence, we failed to find a rigorous mathematical proof of this fact. 5.3. Uniform attractorsWe now turn to the alternative approach to attractors for non-autonomous equations, which is based on their reduction to autonomous ones. We start with a cocycle $\mathcal S_\xi(t)\colon\Phi\to\Phi$, over $T(h)\colon\Psi\to\Psi$, where $\Phi$ and $\Psi$ are Hausdorff topological spaces, and the corresponding family $U_\xi(t,\tau)$, $t\geqslant \tau$, $\xi\in\Psi$, of dynamical processes. Consider the extended phase space $\mathbb P:=\Phi\times\Psi$ and the associated extended dynamical system $\mathbb S(t)$, $t\geqslant 0$, on it defined by
$$
\begin{equation}
\mathbb S(t)(u_0,\xi):=(\mathcal S_\xi(t)u_0,T(t)\xi),\qquad u_0\in\Phi,\quad \xi\in\Psi.
\end{equation}
\tag{5.27}
$$
Indeed, it follows from the cocycle property that $\mathbb S(t)$ is a semigroup. Therefore, we end up with the autonomous dynamical system acting on the extended phase space $\mathbb P$ and may speak about its attractors. To this end, we need to fix a bornology $\mathbb B$ on the space $\Phi$ (in contrast to the previous cases, $B\in \mathbb B$ are time-independent sets $B\subset\Phi$) and to define the extended bornology $\mathbb B_{\rm ext}$ on $\mathbb P$ via $\mathfrak B\subset \mathbb B_{\rm ext}$ if $\Pi_1\mathfrak B\in\mathbb B$ where $\Pi_1\colon\mathbb P\to\Phi$ is a projection onto the first component of the Cartesian product. Definition 5.21. A set $\mathcal B\subset\Phi$ is a uniformly attracting set for the cocycle $\mathcal S_\xi(t)$, $\xi\in\Psi$, with respect to the bornology $\mathbb B$ if for any $B\in\mathbb B$ and any neighbourhood $\mathcal O(\mathcal B)$ of $\mathcal B$ in $\Phi$, there exists $T=T(\mathcal O,B)$ such that It is not difficult to see that if the set $\mathcal B$ is a uniformly attracting set for the cocycle $\mathcal S_\xi(t)$, then the set $\mathcal B_{\rm ext}:=\mathcal B\times\Psi$ is an attracting set for the extended dynamical system $\mathbb S(t)\colon\mathbb P\to\mathbb P$ with respect to the bornology $\mathbb B_{\rm ext}$. Vice versa, if $\mathfrak B$ is an attracting set for $\mathbb S(t)$, then $\mathcal B:=\Pi_1\mathfrak B$ is a uniformly attracting set for the cocycle $\mathcal S_\xi(t)$, $\xi\in\Psi$. Definition 5.22. Let $\Phi$ be a Hausdorff topological space and let $\mathcal S_\xi(t)\colon\Phi\to\Phi$ be a cocycle over the dynamical system $T(h)\colon\Psi\to\Psi$, $h\in\mathbb{R}$. Then the set $\mathcal A_{\rm un}\subset \Phi$ is a uniform attractor for this cocycle with respect to some bornology $\mathbb B$ if 1) $\mathcal A_{\rm un}$ is a compact set in $\Phi$; 2) $\mathcal A_{\rm un}$ is a uniformly attracting set for the cocycle $\mathcal S_\xi(t)$; 3) $\mathcal A_{\rm un}$ is a minimal (by inclusion) set which has properties 1) and 2). An analogue of the existence theorem for attractors in this case reads as follows. Theorem 5.23. Let $\Phi$ and $\Psi$ be Hausdorff topological spaces, and let $\Psi$ be a compact space. In addition, let $\mathcal S_\xi(t)\colon\Phi\to\Phi$ be a cocycle over the group $T(h)\colon\Psi\to\Psi$, $h\in\mathbb{R}$, and let this cocycle possess a compact uniformly attracting set $\mathcal B$ with respect to some bornology $\mathbb B$ on $\Phi$. Then the extended semigroup $\mathbb S(t)\colon\mathbb P\to\mathbb P$ possesses an attractor $\mathbb A\subset\mathcal B\times\Psi$. Moreover, its projection $\mathcal A_{\rm un}:=\Pi_1\mathbb A$ is a uniform attractor for the cocycle $\mathcal S_\xi(t)$. Assume, in addition, that the map $(u_0,\xi)\to(\mathcal S_\xi(t)u_0,T(t)\xi)$ is continuous for every fixed $t$. Then the attractor $\mathbb A$ is strictly invariant with respect to $\mathbb S(t)$. If also $\mathcal B\in \mathbb B$ and $\mathbb B$ is stable with respect to inclusions, then $\mathcal A_{\rm un}$ is generated by all complete $\mathbb B$-bounded trajectories of $\mathcal S_\xi(t)$, that is, where $\mathcal K_\xi$ is a $\mathbb B$-kernel of the dynamical process $U_\xi(t,\tau)$. In particular, since the pullback attractor $\mathcal A_{\xi,{\rm pb}}(\tau)$ of the dynamical process $U_\xi(t,\tau)$ is equal to $\mathcal K_\xi\big|_{t=\tau}$, the relation holds. Sketch of the proof. All the statements of the theorem are straightforward corollaries of our key result, Theorem 3.3, and Propositions 3.4 and 3.6. Indeed, if $\mathcal B$ is a uniformly attracting set for the cocycle $\mathcal S_\xi(t)$, then $\mathfrak B:=\mathcal B\times\Psi$ is a compact attracting set for the extended semigroup $\mathbb S(t)\colon\mathbb P\to\mathbb P$. Thus, the existence of the attractor $\mathbb A$ is verified. The fact that $\mathcal A_{\rm un}=\Pi_1\mathbb A$ is the desired uniform attractor is also straightforward. Indeed, compactness and the attraction property are obvious and we only need to check minimality. Let $\mathcal B_1$ be another uniformly attracting set; then by the minimality of $\mathbb A$, we have $\mathbb A\subset\mathcal B_1\times\Psi$ and $\mathcal A_{\rm un}\subset \mathcal B_1$. The remaining statements are immediate corollaries of Propositions 3.4 and 3.6. Note that the definition of a uniform attractor given above does not require any topology on the space $\Psi$. Moreover, its existence can be obtained on the basis of the existence of a uniformly attracting set, namely, as in the autonomous case, we may define the uniform $\omega$-limit set
$$
\begin{equation}
\omega_{\rm un}(B):=\bigcap_{T\geqslant 0}\biggl[\,\bigcup_{\xi\in\Psi} \bigcup_{t\geqslant T}\mathcal S_\xi(t)B\biggr]_{\Phi}
\end{equation}
\tag{5.29}
$$
and construct the uniform attractor via $\mathcal A_{\rm un}:= \Bigl[\,\bigcup\limits_{B\in\mathbb B}\omega_{\rm un}(B)\Bigr]_{\Phi}$ (see [38] for the details). Then the compactness of $\Psi$ (as well as the topology on it) is not necessary to get a uniform attractor. In particular, we even may construct a uniform attractor for a single dynamical process $U(t,\tau)\colon\Phi\to\Phi$ by introducing the group of shifts $T(h)\colon\Psi\to\Psi$ acting on the space $\Psi=\mathbb{R}$ via $T(h)\xi:=\xi+h$ and by defining the corresponding cocycle $\mathcal S_\xi(t):=U(t+\xi,\xi)$. This corresponds to the trivial reduction of a non-autonomous equation $y'=f(t,y)$ to the autonomous system
$$
\begin{equation*}
\begin{cases} \dot y=f(t,y), \\ \dot t=1. \end{cases}
\end{equation*}
\notag
$$
The drawback of this approach is that it does not give any information on the structure of the uniform attractor (for example, the representation formula (5.28) fails without the compactness of $\Psi$) and it is not clear how to relate the attractor $\mathcal A_{\rm un}$ with the solutions of the PDE under consideration. Thus, if we want to have the representation formula (5.28), we need to consider not only all time shifts of our original equation, but also their limits in the appropriate topology. The next proposition shows that, under natural assumptions, taking this closure does not affect the size of the attractor. Proposition 5.24. Let $\Phi$ and $\Psi$ be two Hausdorff topological spaces, and let $\mathcal S_\xi(t)\colon\Phi\to\Phi$ be a cocycle over a dynamical system $T(h)\colon\Psi\to\Psi$. Assume also that $\Psi_0\subset\Psi$ is a dense set invariant with respect to $T(h)$, $h\in\mathbb{R}$, and that the map $\xi\to \mathcal S_\xi(t)u_0$ is continuous for every fixed $t$ and $u_0\in\Phi$. Then the uniform attractors $\mathcal A_{\Psi}$ and $\mathcal A_{\Psi_0}$ of $\mathcal S_\xi(t)$, considered as the cocycles over $T(h)\colon\Psi\to\Psi$ and $T(h)\colon\Psi_0\to\Psi_0$, exist or do not exist simultaneously and coincide: Sketch of the proof. Indeed, it is not difficult to show using continuity that a set $\mathcal B$ is a compact uniformly attracting set for the cocycle $\mathcal S_\xi(t)$, $\xi\in\Psi$, if and only if it is a compact uniformly attracting set for $\mathcal S_\xi(t)$, $\xi\in\Psi_0$, and this gives the desired result (see, for example, [38]). We see that in order to get the key representation formula (5.28), we need to take a closure of time shifts of the dynamical process under consideration in a topology which, on the one hand, makes the closure a compact Hausdorff topological space and, on the other hand, preserves continuity. This is a non-trivial task, which does not always have a positive solution (in that case the representation formula will be lost); see, for example, [205]. We restrict ourselves to an important particular case (well adapted to the study of equations with additive non-autonomous external forces), where this problem possesses a more or less complete solution. Namely, we assume that $\Phi$ is a separable reflexive Banach space and the elements of $\Psi$ which represent the non-autonomous external forces are functions $\xi\colon\mathbb{R}\to H$ with values in another separable reflexive Banach space $H$. Moreover, assume that we start from a given external force $\xi_0$ which satisfies for some $1<p<\infty$. Then, due to the Banach–Alaoglu theorem, any bounded set in $L^p_{\rm loc}(\mathbb{R},H)$ is precompact in the weak topology. Therefore, the hull
$$
\begin{equation}
\mathcal H(\xi_0):=[T(h)\xi_0,\ h\in\mathbb{R}]_{L_{\rm loc}^{p,w}(\mathbb{R},H)}
\end{equation}
\tag{5.32}
$$
is a compact subset of $L^{p,w}_{\rm loc}(\mathbb{R},H)$, so we may naturally take $\Psi=\mathcal H(\xi_0)$. The scheme works as follows: we start with equation (5.11) with a given non-autonomous external force $\xi_0\in L^p_b(\mathbb{R},H)$ and consider the whole family of similar problems generated by its shifts in time and appropriate limits. Namely, we consider the family
$$
\begin{equation}
\partial_t u=A(u,\xi(t)),\quad \xi\in\Psi:=\mathcal H(\xi_0),\quad u\big|_{t=0}=u_0\in\Phi.
\end{equation}
\tag{5.33}
$$
If these equations are uniquely solvable in an appropriate sense, then they define a cocycle $\mathcal S_\xi(t)\colon\Phi\to\Phi$ over the group $T(h)\colon\Psi\to\Psi$ of time shifts. We fix weak topologies on both spaces $\Phi$ and $\Psi$ and let the bornology $\mathbb B$ on $\Phi$ consist of all bounded subsets of the Banach space $\Phi$. Since $\Phi$ is reflexive, any $B\in\mathbb B$ is precompact in the weak topology, so if we find a uniformly absorbing/attracting set $\mathcal B\in\mathbb B$ for the cocycle $\mathcal S_\xi(t)$ under consideration, then its closed convex hull will be a compact bounded uniformly absorbing set. This, together with Theorem 5.23, gives us the following result. Corollary 5.25. Let the above assumptions hold, and let the cocycle $\mathcal S_\xi(t)\colon\Phi\to\Phi$ over $T(h)\colon\Psi\to\Psi$ with $\Psi=\mathcal H(\xi_0)$ possess a bounded uniformly absorbing set. Assume also that the map $(u_0,\xi)\to\mathcal S_\xi(t)u_0$ is continuous for every fixed $t\geqslant 0$. Then this cocycle possesses a uniform attractor $\mathcal A_{\rm un}^w$ in a weak topology of $\Phi$, and this attractors can be described as follows: where $\mathcal K_\xi$ is a bounded kernel of the dynamical process $U_\xi(t,\tau)$. Note that, as usual, the continuity assumption can be replaced by the closed graph assumption. We now turn to the case of strong attractors, so we want to fix a strong topology on the Banach space $\Phi$. Then, according to the general theory, the existence of a uniform attractor will be guaranteed if we find a compact (in the strong topology) uniformly attracting set for the cocycle $\mathcal S_\xi(t)$ associated with (5.33). As elementary examples show, assumption (5.31) is usually not enough to get this compactness, so some extra assumptions are needed (see [244]). The most natural way would be to assume that the hull $\mathcal H(\xi_0)$ is compact not only in the weak topology of $L^p_{\rm loc}(\mathbb{R},H)$, but also in the strong topology. The functions $\xi_0\in L^p_b(\mathbb{R},H)$ which satisfy this extra assumption are called translation-compact ($\xi_0\in L^p_{\rm\text{tr-c}}(\mathbb{R},H)$). Then $\Psi$ is compact in the strong topology as well and we can fix the strong topology in both $\Phi$ and $\Psi$ and get a complete analogue of Corollary 5.25 for the case of strong topology as well. This scheme, which works for many important dissipative PDEs, was studied in details in [38] (see also [36], [179]), so we do not give further details here. Instead, we discuss a bit more delicate case where we take the strong topology on $\Phi$, but try to keep the weak topology on the hull $\Psi=\mathcal H(\xi_0)$. The possibility to find strong uniform attractors for non-translation-compact external forces was indicated in [159] (see also [157], [158], [160]–[162], [240], [244]). Then we still have an analogue of Corollary 5.25, since we take the strong and weak topologies on the first and second components of the Cartesian product $\mathbb P:=\Phi\times\mathcal H(\xi_0)$, so Theorem 5.23 is still applicable if we have a compact uniformly attracting set for the associated cocycle. Moreover, it is not difficult to show that in this case the uniform attractor in the strong topology coincides with the already constructed $\mathcal A_{\rm un}^w$: and we may use (5.34) to describe the structure of the strong attractor $\mathcal A_{\rm un}^s$. Note that we need not verify the continuity of the maps $(\xi,u_0)\to\mathcal S_\xi(t)u_0)$ in the weak topology on $\Psi$ and the strong topology on $\Phi$ (which is usually not true) and may check continuity (or the closedness of the graph) only in the weak topologies; see [244] for more details.To continue, we need to introduce some classes of external forces. Definition 5.26. Let $\xi_0\in L^p_b(\mathbb{R},H)$, where $H$ is a reflexive Banach space and $1<p<\infty$. We say that $\xi_0$ is time-regular if there exists a sequence $\xi_n\in C^1_b(\mathbb{R},H)$ such that We denote the class of such functions by $L^2_{\rm t\text{-reg}}(\mathbb{R},H)$. Analogously, we say that $\xi_0$ is space-regular, if there exist a sequence of finite-dimensional Banach spaces $H_n\subset H$ and a sequence of functions $\xi_n\in L^2_b(\mathbb{R},H_n)$ such that (5.35) holds. The class of such functions is denoted by $L^p_{\rm\text{sp-reg}}(\mathbb{R},H)$. The function $\xi_0\in L^p_b(\mathbb{R},H)$ is called normal if The class of such functions is denoted by $L^p_{\rm norm}(\mathbb{R},H)$.Remark 5.27. It was shown in [244] that and both spaces on the right-hand side are strictly larger than the space of translation-compact external forces. The machinery for obtaining uniform attractors in a strong topology for various PDEs (including damped wave equations, reaction-diffusion equations, and so on) with non-autonomous external forces from these classes is also presented there. Note that the assumption $1<p<\infty$ is crucial for the theory since it guarantees the reflexivity of the space $L^p_{\rm loc}(\mathbb{R},H)$ and the possibility to use the Banach–Alaoglu theorem. However, the case $p=1$ is also interesting from the point of view of applications, especially for wave equations where it is naturally related to Strichartz’s estimates (see [18], [23], [115], [234]). The situation here is much more delicate since, starting from a regular function $\xi_0\in L^1_b(\mathbb{R},H)$, we may easily get an $H$-valued measure when taking the closure in an appropriate topology. In turn, this leads in a natural way to a dynamical process with trajectories wich are not continuous in time (see [205]. The theory developed there is based on presenting a function $\xi_0\in L^1_b(\mathbb{R},H)$ as a regular Borel measure $\xi_0\in M_b(\mathbb{R},H)$. Then, using the fact that $M(0,1;H)=[C(0,1;H)]^*$, we may endow the space $M_b(\mathbb{R},H)$ with the local $w^*$-topology and consider the hull $\mathcal H(\xi_0)$ in this topology. This is the way how to restore the compactness of the hull, but as a price to pay, we may lose the continuity of the map $\xi\to\mathcal S_\xi(t)$ (see [205] for more details). We illustrate the theory by the example of the 2D Navier–Stokes system with non-autonomous normal external forces. Example 5.28. Consider the system It is also well known that, in contrast to the 3D case, in the 2D case, a weak energy solution is unique and satisfies the energy identity
$$
\begin{equation}
\frac{1}{2}\frac{d}{dt}\|u(t)\|^2_\Phi+ \nu\|\nabla_x u(t)\|^2_{L^2}=(\xi_0(t),u(t))
\end{equation}
\tag{5.37}
$$
(see [12], [215] for the details). This identity gives us the dissipative estimate
$$
\begin{equation}
\|u(t)\|^2_\Phi+\nu\int_0^te^{-\beta(t-s)}\|\nabla_x u(s)\|^2\,ds\leqslant \|u(0)\|_{\Phi}^2\,e^{-\beta t}+C\|\xi_0\|^2_{L^2_b(\mathbb{R},V^{-1})},
\end{equation}
\tag{5.38}
$$
for some positive constants $\beta$ and $C$. We now consider the hull $\Psi=\mathcal H(\xi_0)$ of the external force $\xi_0$ in $L^{2,w}_{\rm loc}(\mathbb{R},H)$, and for every $\xi\in\Psi$ we define a map $\mathcal S_\xi(t)\colon\Phi\to\Phi$ as a solution operator at time $t$ for problem (5.36), where $\xi_0$ is replaced by $\xi$. Obviously, $\mathcal S_\xi(t)$ is a cocycle over $T(h)\colon\Psi\to\Psi$, and the symbol space $\Psi$ is compact if we fix a weak topology on it. We also fix the standard bornology $\mathbb B$ which consists of all bounded subsets of the Banach space $\Phi$. Moreover, it is not difficult to see that estimate (5.38) is uniform with respect to $\xi\in\mathcal H(\xi_0)$ (see, for example, [38]), so by the Banach–Alaoglu theorem the closed ball $\mathcal B_R:=\{u_0\in\Phi\colon \|u_0\|_\Phi\leqslant R\}\in\mathbb B$ is a compact uniformly absorbing set for this cocycle if $R$ is large enough. The continuity of the map $(\xi,u_0)\to \mathcal S_\xi(t)u_0$ in the chosen weak topologies on $\Phi$ and $\Psi$ is also straightforward and, due to Corollary 5.25, this cocycle possesses a uniform attractor $\mathcal A_{\rm un}^w$, which enjoys the representation formula (5.34). We now turn to the case of strong topology in $\Phi$. Since the existence of a uniform attractor $\mathcal A_{\rm un}^w$ in the weak topology of $\Phi$ together with the representation formula has already been established, we only need to find a compact (in the strong topology of $\Phi$) uniformly attracting (or even absorbing) set for this cocycle. We assume, in addition, that $\xi_0\in L^2_{\rm norm}(\mathbb{R},V^{-1})$ and claim that the set is such a set. Indeed, from estimate (5.38) we see that $\mathcal B$ is a bounded uniformly absorbing set, so we only need to verify compactness. To this end we use the energy method. Namely, we consider arbitrary sequences $\xi_n\in\Psi$ and $u_0^n\in\mathcal B_R$ and the sequence of the corresponding solutions $u_n(t):=\mathcal S_{\xi_n}(t)u_n$. Without loss of generality we may assume that $\xi_n\to\xi$ and $u_0^n\to u_0$ in the weak topology and, due to the weak continuity of $\mathcal S_\xi(t)$, we conclude that $u_n(1)\rightharpoondown u(1)$ where $u(t)=\mathcal S_\xi(t)u_0$. Thus, we only need to prove that $u_n(1)\to u(1)$ in the strong topology of $\Phi$. In turn, this will be proved once we have checked that $\|u_n(1)\|_\Phi^2\to\|u(1)\|^2_\Phi$. To get this convergence we rewrite the energy identity (5.37) in the form
$$
\begin{equation*}
\begin{aligned} \, &\frac{d} {dt}(t\|u_n(t)\|^2_\Phi)+N(t\|u_n(t)\|^2_{\Phi})+ 2\nu t\|\nabla_x u_n(t)\|^2_{L^2} \\ &\qquad=(Nt+1)\|u_n(t)\|^2_\Phi+2t(u_n(t),\xi_n(t)), \end{aligned}
\end{equation*}
\notag
$$
where $N$ is an arbitrary positive number, multiply it by $e^{Nt}$, and integrate over $t\in[0,1]$ to get the integral identity
$$
\begin{equation}
\begin{aligned} \, \notag &\|u_n(1)\|^2_\Phi+2\nu\int_0^1e^{-N(1-t)}t\|\nabla_x u_n(s)\|^2_{L^2}\,ds \\ &\qquad=\int_0^1e^{-N(1-t)}(Nt+1)\|u_n(t)\|^2_\Phi\,ds+ 2\int_0^1e^{-N(1-t)}t(u_n(t),\xi_n(t))\,ds. \end{aligned}
\end{equation}
\tag{5.40}
$$
We want to pass to the limit $n\to\infty$ in (5.40). Using the compactness lemma and arguing in a standard way (see, for example, [244]), we conclude that $u_n\to u$ strongly in $L^2(0,1;\Phi)$, so the passing to the limit in the first term on the right-hand side is straightforward. To pass to the limit in the second term on the left-hand side we use the weak lower semicontinuity of convex functions, so it only remains to estimate the last term on the right-hand side. To this end, we use the key property of normal functions, namely, that
$$
\begin{equation*}
\lim_{N\to\infty}\,\sup_{\xi\in\mathcal H(\xi_0)}\,\sup_{t\in\mathbb{R}} \int_0^t e^{-N(t-s)}\|\xi(s)\|_{V^{-1}}^2\,ds=0
\end{equation*}
\notag
$$
(see, for example, [244]). Using this fact in combination with the uniform boundedness of $u_n$ in $L^2(0,1;V)$, we see that, for every $\varepsilon>0$ there exists $N=N(\varepsilon)$ such that
$$
\begin{equation*}
\biggl|\int_0^1e^{-N(1-t)}(u_n(t),\xi_n(t))\,dt\biggr|\leqslant \varepsilon
\end{equation*}
\notag
$$
and the same is true for the limit functions $u$ and $\xi$. Now, passing to the limit $n\to\infty$ in (5.40), we get
$$
\begin{equation}
\begin{aligned} \, \notag &\limsup_{n\to\infty}\|u_n(1)\|^2_\Phi+2\nu\int_0^1e^{-N(1-t)}t \|\nabla_x u(s)\|^2_{L^2}\,ds \\ &\qquad\leqslant \int_0^1e^{-N(1-t)}t(Nt+1)\|u(t)\|^2_\Phi\,ds+2\varepsilon \end{aligned}
\end{equation}
\tag{5.41}
$$
and the comparison with the analogue of (5.40) for the limit functions $u$ and $\xi$ gives where the first inequality is again a weak lower semicontinuity for a convex function. Finally, passing to the limit $\varepsilon\to0$, we arrive at which finishes the proof that $\mathcal B$ is compact. A general theory now gives the existence of a uniform attractor $\mathcal A^s_{\rm un}$ in the strong topology of $\Phi$ and its coincidence with $\mathcal A^w_{\rm un}$. Remark 5.29. It is straightforward to see that $L^2_{\rm t\text{-reg}}(\mathbb{R},H)\subset L^2_{\rm norm}(\mathbb{R},H)$, so the result obtained immediately gives the existence of a strong uniform attractor in the case of time-regular external forces, but gives nothing for space-regular ones. The concept of a normal function can be weakened (following [157], see also [244]), namely, a function $\xi_0\in L^p_b(\mathbb{R},H)$ is weakly normal ($\xi_0\in L^p_{\text{w-norm}}(\mathbb{R},H)$) if for every $\varepsilon>0$ there exist a finite-dimensional subspace $H_\varepsilon\subset H$ and a function $\xi_\varepsilon\in L^p_b(\mathbb{R},H_\varepsilon)$ such that We however note that the class of normal external forces is mainly adapted to parabolic equations and, in contrast to space or time regularity, the normality of the external forces in an appropriate space is not sufficient to have a strong uniform attractor for, say, damped wave equations (see [244]). Remark 5.30. To conclude this section, we mention briefly that the general results of Theorems 3.3 and 5.6 are also well adapted to developing the theory of trajectory attractors for problems in the form (5.33) without the uniqueness of solutions. The detailed exposition of this topic can be found in [38], so we just explain schematically the main ideas. To this end, we introduce, for every $\xi\in\mathcal H(\xi_0)$, the corresponding set $\mathcal K^+_\xi$ of solutions of (5.33) defined on a semi-interval $\mathbb{R}_+$. Analogously to the autonomous case, this may be the set of all weak solutions or some subset of it consisting of some special solutions, but we need to satisfy the key invariance assumption: Alternatively, we may consider maps $T(h)\colon\mathcal K_\xi^+\to\mathcal K_{T(h)\xi}^+$, $h\geqslant 0$, as a cocycle over the dynamical system $T(h)\colon\mathcal H(\xi_0)\to\mathcal H(\xi_0)$, $h\in\mathbb{R}$. Then, fixing some topologies on the spaces $\mathcal K^+_\xi$, we get a family of dynamical processes $U_\xi(t,\tau):=T(t-\tau)\colon\mathcal K^+_{T(\tau)\xi}\to \mathcal K^+_{T(t)\xi}$ and may use Theorem 5.6, our key result, in order to construct the trajectory analogues of pullback attractors (see [246] for more details). 6. Dimensions of the attractorIn this section we start to discuss the finite-dimensionality of attractors related to dissipative PDEs. Note from the very beginning that attractors are usually not regular, but fractal subsets of the phase space, so we need to use generalizations of dimension which are suitable for fractal sets. Actually there are many such generalizations like Lebesgue covering dimension, Hausdorff, fractal, Lyapunov, and Assaud dimensions, and so on. In general, all of them may be different (see, for instance, [64], [193] and also Example 2.6). One of the main motivations to study the dimensions of attractors is to give a rigorous justification of the heuristic idea that, despite the infinite-dimensionality of the initial phase space, the limit dynamics of many important dissipative PDEs is essentially finite-dimensional and can be described by means of finitely many parameters (the order parameters in the terminology of Prigogine, see [187]) whose evolution is governed by a system of ODEs. This finite-dimensional reduction would allow us to reduce the study, say, of turbulence which is described by Navier–Stokes equations to the study of a system of ODEs which can be further investigated using the methods of classical dynamics. Unfortunately, despite many efforts in this direction, the finite-dimensional reduction mentioned above remains a “mystery” and the existing theory is still far from being complete (see [64], [243], [193], and the references therein for more details). 6.1. Mané’s projection theorem and finite-dimensional reductionIn this subsection we restrict ourselves to the best studied case of fractal dimension. Definition 6.1. Let $\mathcal A$ be a (pre)compact set in a metric space $\Phi$. Then, by Hausdorff’s criterion, for any $\varepsilon>0$ it can be covered by finitely many balls of radius $\varepsilon$ in $\Phi$. Let $N_\varepsilon(\mathcal A,\Phi)$ be the smallest number of such balls. Then the Kolmogorov entropy of $\mathcal A$ in $\Phi$ is the following quantity: The applications of the fractal dimension to the above-mentioned finite-dimensional reduction problem are based on the following Mané projection theorem (see [169] and also [106], [193], and the references therein). Theorem 6.2. Let $\mathcal A$ be a compact subset of a Hilbert space $H$ such that $\dim_{\rm f}(\mathcal A,H)\leqslant n$ for some $n\in\mathbb N$. Then the orthoprojector $P_{L}$ onto a “generic” plane $L\subset H$ of dimension $\dim L\geqslant 2n+1$ is one-to-one on $\mathcal A$. Thus, since $\mathcal A$ is compact, $P_{L}\colon\mathcal A\to\overline {\mathcal A}:= P_{L}\mathcal A\subset L$ is a homeomorphism. If $\mathcal A$ is an attractor of a continuous dynamical system $S(t)\colon H\to H$, then we may project this semigroup onto the one acting on a finite-dimensional space $L\cong\mathbb{R}^{2n+1}$:
$$
\begin{equation}
\overline S(t)\colon\overline{\mathcal A}\to\overline{\mathcal A},\quad \overline S(t):=P_{L}\circ S(t)\circ P_{L}^{-1},
\end{equation}
\tag{6.2}
$$
and therefore the dynamics on $\mathcal A$ is indeed described by a continuous semigroup acting on a compact subset of $\mathbb{R}^{2n+1}$. This is exactly the way how we may realize the desired finite-dimensional reduction on the basis on the finiteness of the fractal dimension of the attractor and Mané’s projection theorem. Moreover, we may even write out a system of ODEs on the order parameters $y(t):=P_Lu(t)$ if $u(t)\in\mathcal A$ solves a general PDE of the form Namely, applying the projector $P_{L}$ to both sides of this equation, we formally get
$$
\begin{equation}
\frac{d}{dt}y(t)=P_LA(P_{L}^{-1}y(t))=:\mathcal F(y(t)),\qquad y(t)\in\overline{\mathcal A}\subset L\cong\mathbb{R}^{2n+1}.
\end{equation}
\tag{6.3}
$$
This ODE is often referred to as an inertial form of the initial PDE. Thus, under this approach, the fractal dimension of the attractor $\mathcal A$ is interpreted as the number of effective degrees of freedom in the reduced inertial form. This, in turn, motivates the great interest to various methods for obtaining upper and lower bounds for this dimension (see [12], [179], [215], and the references therein). Remark 6.3. There are many versions of Mané’s projection theorem, in particular, the Hölder Mané projection theorem, which allows us to establish the Hölder continuity of the inverse map $P_{L}^{-1}\colon\overline{\mathcal A}\to\mathcal A$. Moreover, under some extra conditions, which are usually satisfied at least in the case of semilinear parabolic equations, the Hölder exponent can be made arbitrarily close to $1$ by increasing the dimension of $L$. The fact that $H$ is Hilbert is also not essential, and all these results remain true in Banach spaces (see [193] and the references therein). However, the finiteness of the fractal dimension is not enough to guarantee that it is possible to find $L$ in such a way that $P_{L}^{-1}$ is Lipschitz, so in general the inertial form constructed in such a way has only a Hölder continuous vector field $\mathcal F$, and this is the key drawback of the described approach. Indeed, the Hölder continuity of $\mathcal F(y)$ is not enough even to establish the uniqueness of solutions for the inertial form (6.3), so it is not clear how to select the “physically relevant” solutions of this system without referring to the initial PDE. In addition, there are ever more examples where the dynamics on the attractor $\mathcal A$ remains, in a sense, infinite-dimensional (for example, admits super-exponentially stable limit cycles, decaying travelling waves in the Fourier space, and so on) despite the finiteness of the fractal dimension. These attractors cannot be embedded in any finite-dimensional Lipschitz or log-Lipschitz submanifold of the phase space (see [64], [133], [243] and also Section 7 for more details). Thus, despite the widely accepted paradigm, the finite-dimensional reduction based on Mané’s projection theorem does not look as an appropriate solution of the problem and the fractal dimension of the attractor is not an appropriate tool for estimating the effective number of degrees of freedom of the reduced system. We will discuss the alternative methods in the next sections and the rest of this section is devoted to upper and lower bounds for the fractal dimension. 6.2. Upper bounds via the squeezing property: the autonomous caseAssume that the attractor $\mathcal A\subset\Phi$ of the dynamical system under consideration has already been constructed and the maps $S(t)$ are continuous. Then, for every $t_0\in\mathbb{R}_+$, we have and the key question is: under what assumptions on the map $S$ can we guarantee that $\mathcal A$ has a finite fractal dimension? The next theorem gives the simplest of such assumptions.Theorem 6.4. Let $\Phi$ and $\Phi_1$ be two Banach spaces, and let the embedding $\Phi_1\subset\Phi$ be compact. Also assume that $\mathcal A$ is a bounded subset of $\Phi_1$ and the map $S\colon\mathcal A\to\mathcal A$ satisfies (6.4) and the following squeezing/smoothing property: Then the fractal dimension of $\mathcal A$ in $\Phi_1$ is finite and enjoys the following estimate: where $\mathbb H_{1/(4L)}(\Phi_1\hookrightarrow\Phi)$ is the entropy of the unit ball of the space $\Phi_1$ considered as a precompact set in $\Phi$. Proof. Indeed, assume that we have already constructed an $\varepsilon$-net $\{u_k\}_{k=1}^{N(\varepsilon)}$ of the set $\mathcal A$ in $\Phi_1$ (that is, the $\varepsilon$-balls of $\Phi_1$ with centres in $u_k\in\mathcal A$ cover $\mathcal A$). Let us cover every of these balls by $N$ balls of radius $\varepsilon/(4L)$ in $\Phi$. This is possible due to the compactness of the embedding $\Phi_1\subset\Phi$. Moreover, the number of balls which are necessary to cover every one of the balls does not exceed Crucial for us is that this number is independent of $\varepsilon$ and $u_k$. This gives us a covering of $\mathcal A$ by $\varepsilon/(4L)$-balls in $\Phi$, and the number of these balls does not exceed $NN(\varepsilon)$. Moreover, increasing the radii of the balls by the factor of two, we may assume also that the centres $\{v_k\}_{k=1}^{NN(\varepsilon)}$ of these balls belong to $\mathcal A$.
Then, due to the invariance of $\mathcal A$ and condition (6.5), the $\varepsilon/2$-balls in $\Phi_1$ centred at $\{S(v_k)\}$ cover $\mathcal A$. Thus, starting from an $\varepsilon$-covering of $\mathcal A$ which consists of $N(\varepsilon)$ elements, we end up with a new $\varepsilon/2$-covering, with the number of elements $N(\varepsilon/2)\leqslant N N(\varepsilon)$. Since $\mathcal A$ is bounded, we may start with some $\varepsilon_0=R_0$ such that $\mathcal A\subset B_{R_0}(u_0,\Phi_1)$, and therefore $N(\varepsilon_0)=1$. Iterating the procedure described above, we finally get the $\varepsilon_n:=R_0\, 2^{-n}$-coverings which consist of elements. Thus, and this gives us the desired estimate for the fractal dimension of $\mathcal A$.Remark 6.5. To the best of our knowledge, the key idea used in the proof of this theorem is due to Mallet-Paret (see [165], where he used it for estimating the Hausdorff dimension under the extra assumption that $S$ is a $C^1$-map) and the above theorem was proved by Ladyzhenskaya (see [148]). We present a more or less complete proof here since it is simple and elegant on the one hand and, on the other hand, all the known estimates of the fractal dimension of attractors are based on similar iterative schemes. Example 6.6. We return to Example 2.10 of the 1D semilinear parabolic equation (2.11). The existence of solution semigroup $S(t)$ associated with this equation, as well as the existence of a global attractor $\mathcal A\subset H^1_0(0,\pi)\subset C[0,\pi]$ for this semigroup were already verified in Example 2.10. Moreover, due to the maximum/comparison principle we also know that any complete bounded solution $u(t)$, $t\in\mathbb{R}$, of this equation satisfies Remark 6.7. As we can see from Example 6.6, the squeezing property (6.5) is a straightforward corollary of the parabolic smoothing property for the linear PDE (equation of variations) for the difference of two solutions associated with the original nonlinear PDE. Since such a smoothing property is typical for dissipative PDEs (for non-parabolic equations it should be replaced by an appropriate asymptotic smoothing property discussed below), this explains why, in many cases, we have the finiteness of the fractal dimension for attractors of dissipative PDEs. Note also that the above upper bound (6.8) is sharp with respect to $a\to\infty$ (lower bounds of the same order in $a$ are available). It is remarkable that, in order to get reasonably sharp estimates, one should consider the squeezing property (6.5) on a small time interval $t_0\to0$ as $a\to\infty$. In contrast to this, estimates based on volume contraction method usually work better for large $t_0\to\infty$ (see [12], [215]). We mention also that the key advantage of the squeezing property (6.5) is that, in comparison with volume contraction, it does not require the semigroup $S(t)$ to be differentiable with respect to the initial data. This is crucial for example for applications to singular or/and degenerate PDEs, where such differentiability usually does not take place or is very difficult/impossible to verify. As a price to pay, this method may give essentially worse estimates in comparison with the volume contraction. For instance, for the 2D Navier–Stokes equation in a bounded domain, the best known upper bound for the fractal dimension of the attractor reads where $\nu$ is the kinematic viscosity (see [12], [215], and the references therein) but not even bounds polynomial in $\nu^{-1}$ have been obtained via the squeezing property so far.We now state a natural analogue of the squeezing property (6.5) which, in particular, is suitable for non-parabolic equations. Theorem 6.8. Let $\Phi$ and $\Phi_1$ be two Banach spaces, and let the embedding $\Phi_1\subset\Phi$ be compact. Assume also that $\mathcal A$ is a bounded subset of $\Phi_1$ and the map $S\colon\mathcal A\to\mathcal A$ satisfies (6.4) and the following squeezing/smoothing property: Then the fractal dimension of $\mathcal A$ in $\Phi_1$ is finite and satisfies the following estimate: where $\kappa$, $0\leqslant\kappa<1$, and $L>0$ are some constants. The proof of this theorem repeats almost word by word the arguments given in the proof of the previous theorem. The only difference is that, to be able to work with $\kappa>1/2$, we need to use coverings by sets of diameter less than $\varepsilon$ in the iteration scheme (see [44] for more details). We also mention that nowadays there are a huge amount of various formulations of the squeezing property (which give the finiteness of the fractal dimension of an invariant set) adapted to the concrete classes of problems. The detailed exposition of them is out of scope of this survey, so we refer the interested reader to [68], [179], and the references therein. Example 6.9. Consider the degenerate version of the real Ginzburg–Landau equation However, as shown in [69], this attractor has infinite Hausdorff and fractal dimensions:
$$
\begin{equation*}
\dim_{\rm H}(\mathcal A,\Phi)=\dim_{\rm f}(\mathcal A_\gamma,\Phi)=\infty.
\end{equation*}
\notag
$$
Moreover, the infinite-dimensional family of complete bounded solutions can be constructed almost explicitly on the basis of the finite propagation speed of compactly supported solutions. This infinite-dimensionality can be somehow explained by the fact that the formal “linearization” of (6.12) on a degenerate solution $u=0$ is unstable, so the energy income in the system is possible in the “area” where the equation is degenerate. If the areas where the equation is degenerate or singular and the ones where the energy income may occur are separated, then, typically, the corresponding attractor has a finite fractal dimension (see [69], [70], [177]–[180], [207] for the justification of this heuristic principle for concrete classes of singular/degenerate equations). In particular, if we consider a slightly modified version of equation (6.12),
$$
\begin{equation}
\partial_t u=\Delta_x(u^3)+3u^2-2u-u^3,\quad u\big|_{t=0}=u_0,\quad u\big|_{\partial\Omega}=0,
\end{equation}
\tag{6.13}
$$
then the formal linearization $\partial_t v=-2v$ on $u=0$ is exponentially stable and an appropriate version of the squeezing property gives us the finiteness of the fractal dimension of the attractor $\mathcal A$ (see [69]). 6.3. Upper bounds via the squeezing property: the non-autonomous caseThe standard situation in this case is when we have a cocycle $\mathcal S_\xi(t)\colon\Phi\to\Phi$, $\xi\in\Psi$, over a dynamical system $T(h)\colon\Psi\to\Psi$ and an invariant set $\mathcal A_\xi$ such that
$$
\begin{equation*}
\mathcal S_\xi(1)\mathcal A_\xi=\mathcal A_{T(1)\xi}, \qquad \xi\in \Psi.
\end{equation*}
\notag
$$
Instead of the time step $t=1$, one may take an arbitrary $t_0>0$, but for simplicity we assume that $t_0=1$ here. The non-autonomous analogue of the squeezing property (6.10) reads
$$
\begin{equation}
\|\mathcal S_\xi(1)u_1-\mathcal S_\xi(1)u_2\|_{\Phi_1}\leqslant \kappa\|u_1-u_2\|_{\Phi_1}+L(\xi)\|u_1-u_2\|_{\Phi},
\end{equation}
\tag{6.14}
$$
which holds for all $\xi\in\Psi$ and $u_1,u_2\in\mathcal A_\xi$. For simplicity we take a “deterministic” value of $\kappa\in[0,1)$ (it is independent of $\xi$), but its generalization to the “random” case $\kappa=\kappa(\xi)$ is straightforward. The estimate for the fractal dimension of $\mathcal A_\xi$ can be obtained in the same way as in Theorems 6.4 and 6.8. The only difference is that, instead of iterating a single map $S$, we now need to iterate different maps from the cocycle and use the equality
$$
\begin{equation*}
\mathcal S_{T(-1)\xi}(1)\circ\cdots\circ\mathcal S_{T(n-1)\xi}(1)\circ \mathcal S_{T(-n)\xi}(1)\mathcal A_{T(-n)\xi}=\mathcal A_\xi
\end{equation*}
\notag
$$
for all $\xi\in\Psi$ and $n\in\mathbb N$. This gives the following result (see [71], [211] for more details). Theorem 6.10. Let $\Phi_1\subset\Phi$ be two embedded Banach spaces such that the embedding is compact, and let $\mathcal S_\xi(t)\colon\Phi\to\Phi$, $\xi\in\Psi$, be a cocycle over $T(h)\colon\Psi\to\Psi$. Assume that $\xi\to\mathcal A_\xi\subset\Phi_1$ is a family of bounded strictly invariant sets which satisfy the squeezing property (6.14) for some $\kappa\in[0,1)$. Then the fractal dimensions of $\mathcal A_\xi$ satisfies the following estimate: The most straightforward application of this general theorem is related to the “deterministic” (uniform) case, where the size $R(\xi)$ of the attractors $\mathcal A_\xi$ is uniformly bounded: $R(\xi)\leqslant R_0$ and the expanding factor $L(\xi)\leqslant L$ is also uniformly bounded. This is typical for non-autonomous equations with uniformly in time bounded external forces considered in [71] (see also the references therein). Then we have exactly the same estimate for the dimensions of $\mathcal A_\xi$ as in the autonomous case. Corollary 6.11. Let the assumptions of Theorem 6.10 hold, and let, in addition, $R(\xi)\leqslant R_0$ and $L(\xi)\leqslant L$ for all $\xi\in\Psi$. Then the fractal dimensions of the $\mathcal A_\xi$ are finite and satisfy estimate (6.10) uniformly with respect to $\xi\in\Psi$. The applications of this general theorem to the random case are more delicate and interesting. In this case we have an invariant ergodic measure $\mu$ for the dynamical system $T(h)\colon\Psi\to\Psi$ and assume that the function $\xi\to\mathcal A_\xi$ is $\mu$-measurable. In addition, we assume that the expanding factor $L(\xi)$ is also measurable and the entropy of the embedding $\Phi_1\subset\Phi$ has the estimate
$$
\begin{equation}
\mathbb H_\nu(\Phi_1\hookrightarrow\Phi)\leqslant C\biggl(\frac{1}{\nu}\biggr)^\theta
\end{equation}
\tag{6.16}
$$
for some positive $C$ and $\theta$. Thus assumption is not restrictive since in applications $\Phi$ and $\Phi_1$ are usually Sobolev spaces, for which such an estimate holds. Then we have the following result. Corollary 6.12. Let the assumptions of Theorem 6.10 hold, and let, in addition, the random structure mentioned above be introduced and (6.16) hold. Also assume that the family of attractors $\mathcal A_\xi$ is tempered in $\Phi_1$ and the expanding factor $L(\xi)$ has a finite $\theta$th moment: Then, for almost all values $\xi\in\Psi$, the fractal dimension of $\mathcal A_\xi$ is finite and satisfies the estimate Indeed, since $\mathcal A_\xi$ is tempered, $\displaystyle\lim_{n\to\infty}\dfrac{\log_2R(T(-n)\xi)}{n}=0$. The second multiplier on the right-hand side of (6.15) is estimated using (6.16), (6.17), and Birkhoff’s ergodic theorem. Remark 6.13. It is typical for random dynamical systems that, in order to have the finite-dimensionality of the attractor, we need to verify that some random variable (which is responsible for the expansion rate of the distance between two solutions or Lyapunov exponents) has a finite mean (see [54], [58], [211], and the references therein). This condition is not trivial and is often the most difficult to verify. On the other hand, as we will see in the next toy example (suggested in [30]), the dissipation mechanism may be not strong enough to provide the finite-dimensionality of a random attractor if this condition is violated. Example 6.14. Let $H=l_2$ (the space of square summable sequences), and consider the RDS in $H$ generated by the following equations: The system of ODEs (6.19) can be solved explicitly. In particular, if $aq-(1-q)b>0$, then
$$
\begin{equation}
u_1(t)=u_{1,\gamma}(t)=\int_{-\infty}^t \exp\biggl\{-\int_s^t\gamma(l)\,dl\biggr\}\,ds
\end{equation}
\tag{6.20}
$$
is a unique tempered complete solution of the first equation (for almost all $\gamma\in\Psi$). Moreover, it is not difficult to show that if In this case the following result holds. Proposition 6.15. Let the exponents $a,b>0$ and $q\in(0,1)$ satisfy (6.22). Then the random attractor $\mathcal A_\gamma$ for system (6.19) in $H$ has infinite Hausdorff and fractal dimensions: for almost all $\gamma\in\Psi$. Sketch of the proof. The detailed proof of this result was given in [30]. Here we just discuss briefly the main ideas behind it. The existence of a random tempered absorbing set for the first component $u_1$ of system (6.19) follows from the explicit formula for the solution. Now we assume that $u_1(\tau)$ is already in this absorbing set and get estimates for $u_k$. To this end, multiplying the $k$th equation by $\operatorname{sgn}(u_k)$ and taking the sum, after standard estimates, we get Integrating this inequality and using that $u_1(t)$ is tempered, we get a tempered absorbing ball for $u$ in $l_1\subset H$. To obtain a compact absorbing set it is enough to use the parabolic smoothing property in a standard way. Thus, the existence of a random attractor $\mathcal A_\gamma$ is verified.
Recall also that a random attractor consists of all complete tempered trajectories of the system under consideration, so it is enough to find all such trajectories. The first equation is linear and independent of the other equations, so such a trajectory is unique and is given by (6.20). Thus, to find $\mathcal K_\gamma$, we need to fix $u_1(t)=u_{1,\gamma}(t)$ in the other equations of (6.20) and find the tempered attractor $\mathcal A^k_\gamma$, $k=2,3,\dots$, for every component of (6.20) separately. Then the desired attractor $\mathcal A_\gamma$ for the whole system is presented as a product:
$$
\begin{equation}
\mathcal A_\gamma=\{u_{1,\gamma}(0)\}\times \bigotimes_{k=2}^\infty\mathcal A^k_\gamma.
\end{equation}
\tag{6.24}
$$
Moreover, since the attractor is always connected, every $\mathcal A^k_\gamma$ is a closed interval, so to prove the desired infinite dimensionality, it is enough to prove that $\mathcal A^k_\gamma\ne\{0\}$ for all $k$. In other words, we need to find a non-zero tempered trajectory $u_k=u_k(t)$ for the equation using that $u_{1,\gamma}(t)$ is tempered and has an infinite mean.
To this end, we note first that every solution of equation (6.25) is either tempered as $t\to-\infty$ or blows up backward in time (this can easily be shown by comparison using the fact that $u_{1,\gamma}(t)$ is tempered), so any complete solution $u_k(t)$, $t\in\mathbb{R}$, is automatically tempered. We construct this solution by solving equation (6.25) explicitly. Namely,
$$
\begin{equation*}
u_{k,\gamma}(t):=\biggl(2\int_{-\infty}^t \exp\biggl\{2\int_s^t(k^4-u_{1,\gamma}(l))\,dl\biggr\}\,ds\biggr)^{-1/2}.
\end{equation*}
\notag
$$
Indeed, the finiteness of the integral is guaranteed by (6.21) (see [30] for more details). Thus, we have proved that $\mathcal A^k_\gamma\ne \{0\}$ for all $k$ and almost all $\gamma$ and the proposition is proved. Remark 6.16. Actually, we have found explicitly the random attractor in the previous example: This example shows that, in contradiction with the commonly accepted paradigm, adding random terms may not only simplify the dynamics, but may also make it essentially more complicated and even infinite-dimensional. We expect that this phenomenon has a general nature and can be observed in more realistic equations. We conclude this subsection by considering the applications of the dimension estimate to uniform attractors. In this case we typically have a cocycle $S_\xi(t)\colon\Phi\to\Phi$, $\xi\in\Psi$, over the dynamical system $T(h)\colon\Psi\to\Psi$, a family of invariant sets (pullback attractors) $\mathcal A_\xi$, $\xi\in\Psi$, and a union
$$
\begin{equation}
\mathcal A_{\rm un}=\bigcup_{\xi\in\Psi}\mathcal A_\xi,
\end{equation}
\tag{6.26}
$$
which is exactly the uniform attractor whose dimension we want to estimate. For simplicity we assume that $\Psi\subset L^p_b(\mathbb{R},H)$ for some Banach space $H$ and $1\leqslant p<\infty$ is a hull of some translation-compact external force $\xi_0\in L^p_b(\mathbb{R},H)$, so we assume that
$$
\begin{equation*}
\Psi:=\mathcal H(\xi_0)\Subset L^p_{\rm loc}(\mathbb{R},H)
\end{equation*}
\notag
$$
(see [38] for more details and more general exposition). It is also well known that, despite the finiteness of the fractal dimension of every $\mathcal A_\xi$, the dimension of the union (6.26) is usually infinite, because of the infinite-dimensionality of the hull $\mathcal H(\xi_0)$. For this reason it looks natural (following Vishik and Chepyzhov) to study the Kolmogorov $\varepsilon$-entropy of $\mathcal A_{\rm un}$. To this end we need to add to the squeezing property (6.10) also some kind of Lipschitz/Hölder continuity with respect to $\xi$, namely, to assume that
$$
\begin{equation}
\|\mathcal S_{\xi_1}(1)u_1-\mathcal S_{\xi_2}(1)u_2\|_{\Phi_1}\leqslant \kappa\|u_1-u_2\|_{\Phi_1}+L\|u_1-u_2\|_{\Phi}+K\|\xi_1-\xi_2\|_{L^p(0,1;H)},
\end{equation}
\tag{6.27}
$$
where $0\leqslant\kappa<1$, $L$ and $K$ are independent of $\xi_1,\xi_2\in\Psi$, and $u_1,u_2\in\mathcal A_\xi$. Then we get the following analogue of Theorem 6.8. Theorem 6.17. Let $\mathcal S_\xi(t)\colon\Phi\to\Phi$ be a cocycle over a dynamical system $T(h)\colon\Psi\to\Psi$ and $\Psi=\mathcal H(\xi_0)$ be a hull of some function $\xi_0\in L^p_b(\mathbb{R},H)$ which is translation-compact in this space. Assume also that there is a family of invariant sets $\mathcal A_\xi$ which satisfies the squeezing property (6.27), where $\Phi_1$ is some other Banach space compactly embedded in $\Phi$, and also is uniformly bounded in the Banach space $\Phi_1$. Then the Kolmogorov entropy of the union (6.26) has the following estimate: The derivation of this estimate is similar to the proof of Theorem 6.4, so we omit it here and refer the interested reader to [67] or [234] (see also [38] for the analogous results obtained via the volume contraction method). As we already mentioned, the dimension of $\mathcal A_{\rm un}$ may be infinite due to the second term on the right-hand side of (6.28) (see [38] and the references therein). However, there are interesting particular cases where the dimension of a uniform attractor remains finite; for example, it is so for periodic or quasi-periodic dynamical processes as well as for dynamical processes stabilizing to autonomous ones as time tends to $\pm\infty$ (see [38], [67], [234] for more examples). To extract this finite dimensionality from the key estimate (6.28), we introduce the fractal dimension of the hull $\mathcal H(\xi_0)$ as follows:
$$
\begin{equation}
\dim_{\rm f}(\mathcal H(\xi_0)):=\limsup_{\varepsilon\to0} \frac{\mathbb H_{\varepsilon}(\mathcal H(\xi_0)|_{t\in[0, \log_2(1/\varepsilon)]},L^p_b([0,\log_2(1/\varepsilon)],H))} {\log_2(1/\varepsilon)}\,.
\end{equation}
\tag{6.29}
$$
Then estimate (6.28) reads
$$
\begin{equation}
\dim_{\rm f}(\mathcal A_{\rm un},\Phi_1)\leqslant \frac{\mathbb H_{(1-\kappa)/(4L)}(\Phi_1\hookrightarrow\Phi)} {\log_2(2/(1+\kappa))}+\dim_{\rm f}(\mathcal H(\xi_0))
\end{equation}
\tag{6.30}
$$
(see [38] for details). The first term on the right-hand side can be interpreted as (an upper bound for) the dimension of every $\mathcal A_\xi$ and the second term is added due to the union. Remark 6.18. It is not difficult to see that $\dim_{\rm f}(\mathcal H(\xi_0))$ is indeed the standard fractal dimension of the set $\mathcal H(\xi_0)$ in the weighted norm of $L^p_{e^{-|x|}}(\mathbb{R},H)$. Note that, for a translation-compact $\xi_0\in L^p_b(\mathbb{R},H)$, the hull $\mathcal H(\xi_0)$ is compact in the local topology of $L^p_{\rm loc}(\mathbb{R},H)$ and, in order to speak about the dimension, we need to fix a metric on $\mathcal H(\xi_0)$. Estimate (6.28) hints that the weighted metric of $L^p_{e^{-|x|}}(\mathbb{R},H)$ is the most appropriate one for estimating the entropy of uniform attractors. Note also that estimate (6.28) makes no sense if $\xi_0$ is not translation-compact (since the second term on the right-hand side will be infinite). However, there is a natural generalization of this formula, which was proposed in [234] and allows us to work with the classes of non-translation-compact external forces considered in subsection 5.3. It is worth mentioning that the result of Theorem 6.17 remains true with minor changes if we replace Lipschitz continuity with respect to $\xi$ in (6.27) by Hölder continuity with an arbitrary exponent $\alpha\in(0,1]$. In contrast to this, one may lose finite dimensionality (even in the autonomous case) if one replaces Lipschitz continuity with respect to the initial data $u_i$ by Hölder or $\log$-Lipschitz continuity (see [173] for the counterexample related to attractors of elliptic PDEs). 6.4. Volume contraction and Lyapunov dimensionWe now turn to the scheme for estimating the dimension of the attractor, which is based on volume contraction arguments. This scheme has become extremely popular due to applications to the Navier–Stokes equations, where it gives the best estimates available so far, which cannot be obtained using other methods (although this scheme has significant drawbacks, for example, it requires the phase space to be Hilbert and the corresponding dynamical system to be differentiable with respect to the initial data, so it cannot replace other methods fully). Since there are nice expositions of this material in the literature (see, for example, [12], [215], and the references therein), we restrict ourselves to just a brief discussion. We start with introducing the volume contraction factor. Let $\Phi$ be a separable Hilbert space, and let $\varphi_1,\dots,\varphi_d\in\Phi$. By definition, the wedge product $\varphi_1\wedge\cdots\wedge\varphi_d$ is the $d$-linear antisymmetric form on $\Phi$ defined by
$$
\begin{equation*}
\varphi_1\wedge\cdots\wedge\varphi_d(\psi_1,\dots,\psi_d):= \det\bigl((\varphi_i,\psi_j)_{i,j=1}^d\bigr),
\end{equation*}
\notag
$$
where $(\,\cdot\,{,}\,\cdot\,)$ is the inner product in $\Phi$. A $d$-linear form on $\Phi$ that is a wedge product of $d$ vectors of $\Phi$ is called decomposable. Let us denote by $\tilde\Lambda^d\Phi$ the space of $d$-linear antisymmetric forms which can be presented as a finite linear combination of decomposable functionals. For two decomposable forms $\varphi_1\wedge\cdots\wedge\varphi_d$ and $\psi_1\wedge\cdots\wedge\psi_d$, their inner product is defined by
$$
\begin{equation*}
(\varphi_1\wedge\cdots\wedge\varphi_d, \psi_1\wedge\cdots\wedge\psi_d) := \det\bigl((\varphi_i,\psi_j)_{i,j=1}^d\bigr),
\end{equation*}
\notag
$$
and being extended to $\tilde\Lambda^d\Phi$ by linearity, it defines an inner product on the space $\tilde\Lambda^d\Phi$ (see [215] for the details). Finally, the completion of $\tilde\Lambda^d\Phi$ with respect to this norm is called the $d$th exterior power of the space $\Phi$ and is denoted by $\Lambda^d\Phi$. Note that $\Lambda^d\Phi$ is the subspace of all continuous antisymmetric $d$-linear forms on $\Phi$, and this subspace is proper if $d>1$ and $\dim\Phi=\infty$. Now let $L\in\mathcal L(\Phi,\Phi)$ be a linear continuous operator on $\Phi$. Then its $d$th exterior power $\Lambda^dL$ is defined by
$$
\begin{equation*}
(\Lambda^dL\xi)(\psi_1,\dots,\psi_d):=\xi(L^*\psi_1,\dots,L^*\psi_d),\qquad \xi\in \Lambda^d\Phi,
\end{equation*}
\notag
$$
where $L^*$ is the operator adjoint to $L$. It is not difficult to check that $\Lambda^dL:\Lambda^d\Phi\to\Lambda^d\Phi$ is a linear continuous operator and its norm does not exceed $\|L\|^d$. The following lemma gives a geometric interpretation for the norm of this operator. Lemma 6.19. The norm of the $d$ th exterior power of $L$ satisfies The proof of this lemma can be found, for example, in [215]. Thus, geometrically, $\|\Lambda^dL\|$ is the maximal expanding factor for $d$-dimensional volumes under the action of the operator $L$. Now assume that we are given a compact set $\mathcal A\subset\Phi$ and a map $S\colon\mathcal A\to\mathcal A$. Assume that this map is uniformly Fréchet (quasi)differentiable on $\mathcal A$, that is, there exist linear continuous maps $S'(u_0)\colon\Phi\to\Phi$, $u_0\in\mathcal A$, such that
$$
\begin{equation}
\|S(u_1)-S(u_2)-S'(u_1)(u_1-u_2)\|_{\Phi}=o(\|u_1-u_2\|_{\Phi})
\end{equation}
\tag{6.31}
$$
uniformly with respect to $u_1,u_2\in\mathcal A$. Assume also that the map $u_0\to S'(u_0)$ is continuous on $\mathcal A$. Then we define an infinitesimal $d$-volume contraction factor $\omega_d(S,\mathcal A)$ by
$$
\begin{equation*}
\omega_d(S,\mathcal A):=\sup_{u_0\in\mathcal A} \|\Lambda^dS'(u_0)\|_{\mathcal L(\Lambda^d\Phi,\Lambda^d\Phi)}.
\end{equation*}
\notag
$$
This definition can be extended also to non-integer values of $s=s_0+\alpha$, $\alpha\in(0,1)$, via
$$
\begin{equation*}
\omega_d(S,\mathcal A):=\sup_{u_0\in\mathcal A} \bigl\{\|\Lambda^dS'(u_0)\|_{\mathcal L(\Lambda^d\Phi, \Lambda^d\Phi)}^{1-\alpha} \|\Lambda^{d+1}S'(u_0)\|_{\mathcal L(\Lambda^{d+1} \Phi,\Lambda^{d+1}\Phi)}^{\alpha}\bigr\}.
\end{equation*}
\notag
$$
The main result of the theory is the following theorem. Theorem 6.20. Let $\mathcal A$ be a compact set in a Hilbert space $\Phi$ which is strictly invariant with respect to a map $S\colon\mathcal A\to\mathcal A$. Assume also that the map $S$ is uniformly quasi-differentiable on $\mathcal A$ and the map $u_0\to S'(u_0)$ is continuous. Finally, let $\omega_d(S,\mathcal A)<0$ for some $d\in\mathbb{R}_+$. Then To the best of our knowledge, this theorem has been proved by Douady and Oesterlé [60] in the finite-dimensional case for the Hausdorff dimension and has been extended to the infinite-dimensional case in [107] (see also [49]). The case of fractal dimension is more delicate and had initially been treated with some rather annoying extra conditions (see [49]). These conditions were removed by Hunt [105] in the finite-dimensional case (see also [19] for $C^2$-diffeomorphisms in the infinite-dimensional case). The result of the theorem in the form stated above appeared in [32]. Remark 6.21. The proof of Theorem 6.20 follows the general scheme presented in Theorem 6.4. Namely, we start with the covering of $\mathcal A$ by very small $\varepsilon$-balls and obtain a better covering of the set $S(\mathcal A)$ using our assumptions on $S$. Then we iterate this process in order to get an estimate for the dimension. Indeed, since $\varepsilon$ is small and $S$ is smooth enough, the image $S(B_\varepsilon(u_0,\Phi))$ is close to the ellipsoid $S'(u_0)B_\varepsilon(u_0,\Phi)$, and by the assumptions of the theorem the $d$-dimensional volume of this ellipsoid is strictly less than the $d$-dimensional volume of the initial ball. In the case when the Hausdorff dimension is considered, it is more or less straightforward to cover every such ellipsoid by smaller balls of different radii in such a way that the corresponding $d$-dimensional Hausdorff measure is contracted at every step of iterations, which gives the result (see [215] for the details). The difficulties in the case of fractal dimension are related to the fact that, in contrast to the case of conditions (6.5), different ellipsoids obtained as the images of $\varepsilon$-balls from the initial covering may have considerably different “shapes” (only their volumes are contracted), but we still need to cover them by balls of the same radius, say $\varepsilon/K$. For this reason, much more delicate estimates are required. Note that the quasi-differentiability condition is automatically satisfied if the map $S\colon\Phi\to\Phi$ is $C^1$-smooth. However, in applications it is often easier to verify estimate (6.31) on the attractor only, since the attractor is usually more smooth. This is the reason why quasi-differentiability is introduced. Note also that, in contrast to the Fréchet derivative, the linear operator $S'(u_0)$ satisfying (6.31) may be not unique. It is worth mentioning that, in applications to PDEs, we normally use the theorem for integer values of $d$ only. However, the possibility to use non-integer $d$s is crucial in many applications to ODEs, for example, to the Lorenz system. Theorem 6.20 can be reformulated in a more elegant way using the fact that the map $S$ can be replaced by any iteration $S^n$ of it. Indeed, it is not difficult to show using the chain rule that the sequence $\ln\omega_d(S^n,\mathcal A)$ is subadditive, so the limit
$$
\begin{equation*}
\overline\omega_d(S,\mathcal A)=\lim_{n\to\infty} \bigl(\omega_d(S^n,\mathcal A)\bigr)^{1/n}= \inf_{n\in\mathbb N}\bigl(\omega_d(S^n,\mathcal A)\bigr)^{1/n}
\end{equation*}
\notag
$$
exists for every $d$. In the case of continuous time and a semigroup $S(t)$, we just replace $n\in\mathbb N$ by $t\in\mathbb{R}_+$ and $S^n$ by $S(t)$ throughout. Also recall that the uniform Lyapunov dimension of the map $S$ on $\mathcal A$ is defined by
$$
\begin{equation}
\dim_{\rm L}(S,\mathcal A):=\sup_{d\in\mathbb{R}_+} \{\overline\omega_d(S,\mathcal A)\geqslant 0\}
\end{equation}
\tag{6.32}
$$
(see [215], [242], and the references therein). Then Theorem 6.20 reads as follows. Corollary 6.22. Let the assumptions of Theorem 6.20 hold. Then At the next step we discuss how to evaluate or estimate the Lyapunov dimension. This can be done for dynamical systems with continuous time using analogues of Liouville’s formula for the evolution of volumes. Namely, assume that, for any $v_0\in\Phi$ and any $u_0\in\mathcal A$ the (quasi-)derivative $v(t):=S'(t)(u_0)v_0$ of the map $S(t)\colon\mathcal A\to\mathcal A$ solves the corresponding equation of variations:
$$
\begin{equation}
\partial_t v(t)=\mathcal L(u(t))v(t),\quad v\big|_{t=0}=v_0,\quad u(t):=S(t)u_0,
\end{equation}
\tag{6.34}
$$
where $\mathcal L(u)\colon D\to\Phi$ are linear unbounded operators such that $D$ is dense in $\Phi$ and the operators $\mathcal L(u)$ are bounded above, that is,
$$
\begin{equation*}
(\mathcal L(u)\eta,\eta)\leqslant C\|\eta\|^2_{\Phi},\qquad \eta\in D,\quad u\in\mathcal A,
\end{equation*}
\notag
$$
where $C$ is independent of $\eta$ and $C$. The following lemma is a key technical tool in estimating the Lyapunov dimension. Lemma 6.23. Under the above assumptions, the following identity holds: This statement is the extension of the classical Liouville theorem to an infinite-dimensional case (see, for example, [215] for the details). To proceed further, we define the $d$-dimensional trace of the operator $\mathcal L(u(t))$, namely,
$$
\begin{equation}
\operatorname{Tr}_d\bigl(\mathcal L(u(t))\bigr):= \sup\biggl\{\,\sum_{i=1}^d(\mathcal L(u(t))\psi_i,\psi_i)\colon \psi_i\in D, \ (\psi_i,\psi_j)=\delta_{ij}\biggr\}.
\end{equation}
\tag{6.36}
$$
Then identity (6.35), together with the obvious estimate
$$
\begin{equation*}
\operatorname{Tr}(Q(t)\circ\mathcal L(u(t))\circ Q(t))\leqslant \operatorname{Tr}_d\bigl(\mathcal L(u(t))\bigr),
\end{equation*}
\notag
$$
gives us the estimate
$$
\begin{equation*}
\bigl(\omega_d(S(T),\mathcal A)\bigr)^{1/T}\leqslant \exp\biggl\{2\sup_{u_0\in\mathcal A}\biggl\{\frac{1}{T}\int_0^T \operatorname{Tr}_d(\mathcal L(u(t)))\,dt\biggr\}\biggr\}.
\end{equation*}
\notag
$$
This, in turn, gives a corollary of Theorem 6.20 which is suitable for applications (see [38], [215] for more details). Corollary 6.24. Let the above assumptions hold. Also let Assume that $\overline q_{d}<0$ for some $d\in\mathbb N$. Then Remark 6.25. Corollary 6.24 reduces the problem of estimating the fractal dimension of the attractor to estimating the traces of linear operators related to the corresponding equation of variations. This links the theory of attractors with a classical topic of operator theory, namely, estimating the traces of various differential operators (see [80] and the references therein). In particular, in relatively simple cases, such estimates can be obtained using the min-max principle, while in more complicated cases (like the Navier–Stokes equations or other hydrodynamical equations) an essential progress has been achieved using the Lieb–Thirring inequalities, as well as other collective Sobolev inequalities (see [110], [109], [215], and the references therein). Note also that, although the definition of the Lyapunov dimension is independent of the concrete choice of the equivalent inner product in the space $\Phi$, a clever choice of this inner product (which may depend on the point of the phase space), may essentially improve the estimates; see, for example, [85] and [204] for estimates related to the damped Schrödinger and hyperbolic Cahn–Hilliard equations, respectively, as well as [151] for the calculation of the exact value of the Lyapunov dimension of the Lorenz attractor. We will not go into further details here and restrict ourselves to an example related to damped wave equations, which is important for the discussion in the next subsection. Example 6.26. Consider an abstract damped wave equation in a Hilbert space $\Phi$: Our main task here is to estimate the fractal dimension of this attractor and its dependence on the small parameter $\gamma$. Since we have made the extremely strong assumptions (6.40) on the nonlinearity $F$, it is also immediate to verify that the solution semigroup $S(t)$ is $C^\infty$-smooth in $E$ with respect to the initial data and the Fréchet derivative $\xi_v(t):=S'(\xi_0)(t)\xi_{v_0}$ is defined as a solution of the following equation of variations:
$$
\begin{equation}
\partial_t^2 v+\gamma\partial_t v+Av=F'_u(\xi_u(t))v+ F'_{\partial_t u}(\xi_u(t))\partial_t v,\quad \xi_v\big|_{t=0}=\xi_{v_0}, \quad \xi_u(t):=S(t)\xi_{u_0},
\end{equation}
\tag{6.41}
$$
so we only need to estimate the $d$-dimensional traces of the operator
$$
\begin{equation}
\mathcal L(\xi_u(t)):=\begin{pmatrix} 0&1 \\ -A+F'_u&-\gamma +F'_{\partial_t u} \end{pmatrix}
\end{equation}
\tag{6.42}
$$
in the space $E$. However, the volume contraction scheme does not work properly for the original inner product in $E$ induced by the Cartesian product and we need to modify it, namely, we set
$$
\begin{equation*}
(\{v,v'\},\{w,w'\})_\gamma:=(Av,w)+(v',w')+ \frac{\gamma}{2}((v,w')+(v',w)),\quad \{v,v'\},\{w,w'\}\in E.
\end{equation*}
\notag
$$
Then this inner product is equivalent to the standard one if $\gamma>0$ is small enough and
$$
\begin{equation*}
\begin{aligned} \, (\mathcal L\xi_v,\xi_v)_\gamma&=-\frac\gamma2\bigl(\|v\|^2_{\Phi^1}+ \|v'\|^2_{\Phi}\bigr)+(F'_{\partial_t u}v',v')+(F'_uv,v') \\ &\qquad+ \frac{\gamma}{2}\bigl((F'_uv,v)+(F'_{\partial_t u}v',v)-\gamma(v,v')\bigr). \end{aligned}
\end{equation*}
\notag
$$
To estimate the terms containing $F$ on the right-hand side, we use assumption (6.40). For instance, for the first term we have
$$
\begin{equation*}
|(F'_{\partial_t u}v',v')|\leqslant \|F'_{\partial_t u}v'\|_{\Phi^m}\|v'\|_{\Phi^{-m}}\leqslant C\|v'\|_{\Phi^{-m}}^2=C(A^{-m}v',v')
\end{equation*}
\notag
$$
and estimating the remaining terms analogously, for sufficiently small $\gamma>0$ we arrive at the inequality
$$
\begin{equation*}
(\mathcal L(\xi_u(t))\xi_v,\xi_v)_\gamma\leqslant (\mathbb L_m\xi_v,\xi_v)_\gamma,\quad \mathbb L_m:=\begin{pmatrix} -\gamma/4+C_mA^{-m}&0 \\ 0&-\gamma/4+C_m A^{-m} \end{pmatrix},
\end{equation*}
\notag
$$
where $m\in\mathbb N$ is arbitrary and the constant $C_m$ is independent of $\xi_v=\{v,v'\}$ and $\xi_u$ (see [220]). Then, using the min-max principle, we finally arrive at
$$
\begin{equation*}
\operatorname{Tr}_d(\mathcal L(\xi_u(t)))\leqslant \operatorname{Tr}_d(\mathbb L_m)=2\biggl(-\frac{\gamma}{4} d+ C_m\sum_{n=1}^d\lambda_n^{-m}\biggr),
\end{equation*}
\notag
$$
where $\{\lambda_n\}_{n=1}^\infty$ are the eigenvalues of the operator $A$ enumerated in the non-decreasing order. If we now assume that for some large $m$ (which is always true if $A$ is a uniformly elliptic differential operator in a bounded domain), estimate (6.37) will show us that and then Corollary 6.24 gives us the final result that the fractal dimension of the attractor $\mathcal A$ associated with equation (6.39) has the following upper bound:
$$
\begin{equation}
\dim_{\rm f}(\mathcal A,E)\leqslant\dim_{\rm L}(S(t),\mathcal A) \leqslant \frac C\gamma,
\end{equation}
\tag{6.43}
$$
where $\gamma$ is small enough and $C$ is independent of $\gamma$ (see [220] for details). 6.5. Lower bounds for the dimensionMost lower bounds for the dimensions of attractors which are available in the literature are based on estimates for the instability index of an appropriately constructed equilibrium. Assume that we are given a dissipative semigroup $S(t)\colon\Phi\to\Phi$ acting on a Banach space $\Phi$ with the standard bornology $\mathbb B$ which consists of all bounded sets of $\Phi$. Assume also that $u_0$ is an equilibrium of this semigroup, and consider the unstable set of this equilibrium:
$$
\begin{equation}
\mathcal M_+(u_0):=\Bigl\{v_0\in\Phi: \text{ there exists } u\in\mathcal K \text{ such that } u(0)=v_0,\ \lim_{t\to-\infty}u(t)=u_0\Bigr\},
\end{equation}
\tag{6.44}
$$
where $\mathcal K$ is the set of complete bounded trajectories of $S(t)$. Note that, due to dissipativity, it is enough to construct a negative semi-trajectory $u(t)$, $t\leqslant0$, with the above properties. Then it follows from the definition of an attractor that if the attractor $\mathcal A$ exists. Thus, the fractal dimension of the attractor $\mathcal A$ is estimated in terms of the dimension of the unstable set $\mathcal M_+(u_0)$, which is usually a submanifold itself or contains a submanifold of sufficiently large dimension. Indeed, assume that $S(t)$ is Fréchet differentiable near the equilibrium point $u_0$ and the Fréchet derivative $S'(u_0)$ is given by the equation of variations (6.34) and is uniformly continuous in a small neighbourhood of $u_0$. Assume also that the part of the spectrum $\sigma(\mathcal L(u_0))$ belonging to the positive semi-plane $\operatorname{Re}\lambda>0$ consists of finitely many eigenvalues of finite algebraic multiplicity. Denote by $\mathcal N_+(u_0)\in\mathbb N$ the number of all such eigenvalues, taking their multiplicities into account. Then the standard theorem about unstable manifolds (see, for example, [12]) claims that the set $\mathcal M_+(u_0)$ contains an $\mathcal N_+(u_0)$-dimensional local submanifold which is generated by all complete trajectories which approach $u_0$ exponentially fast as $t\to-\infty$. Hence
$$
\begin{equation}
\dim_{\rm f}(\mathcal A,\Phi)\geqslant \dim_{\rm f}(\mathcal M_+(u_0),\Phi)\geqslant \mathcal N_+(u_0).
\end{equation}
\tag{6.46}
$$
Thus, obtaining lower bounds for the dimension of an attractor is also reduced to a classical problem in spectral theory, namely, finding/estimating the number of unstable eigenvalues of a given differential operator. Of course, the scheme described above can be used not only in the case where $u_0$ is an equilibrium. It may be something more complicated, for example, a periodic or a quasi-periodic or even a chaotic orbit. The theory of unstable manifolds for such objects is well developed nowadays (see, for example, [12], [116], and the references therein). However, such objects are almost never used in the literature to estimate the dimension of an attractor since it is extremely difficult to find them and estimate explicitly their instability index. An exception from this, which is based on the theory of homoclinic bifurcations and, as believed, has a general nature, will be considered in examples below. We also mention that lower bounds for the Lyapunov dimension of an attractor can be obtained via equilibria and the obvious inequality
$$
\begin{equation*}
\dim_{\rm L}(S(t),\mathcal A)\geqslant \dim_{\rm L}(S(t),u_0).
\end{equation*}
\notag
$$
It is important that the Lyapunov dimension $\dim_{\rm L}(S(t),u_0)$ can also be estimated via the spectrum of the infinitesimal generator $\mathcal L(u_0)$ of the linear semigroup $S'(t)(u_0)$. To this end, we need the following definition. Definition 6.27. Let $\Phi$ be a Hilbert space and $\mathcal L(u_0)\colon D\to\Phi$ be a linear closed bounded above operator in $\Phi$. Also let $\mu_\infty:=\sup\operatorname{Re}\sigma_{\rm ess}(\mathcal L(u_0))$. Then there is only a finite or countable number of eigenvalues $\lambda_j$ of $\mathcal L(u_0)$ in the half-plane $\operatorname{Re}\lambda>\mu_\infty$, which can be enumerated in the non-increasing order. Then we define the Lyapunov exponents $\mu_n$, $n\in\mathbb N$, as follows: $\mu_n=\operatorname{Re}\lambda_n$ if there exists at least $n$ eigenvalues satisfying $\operatorname{Re}\lambda>\mu_\infty$, and $\mu_n=\mu_\infty$ otherwise. Then the Lyapunov dimension of $\mathcal L(u_0)$ is defined by The next proposition gives us the desired estimate. Proposition 6.28. Let the assumptions of Theorem 6.20 hold, and let $u_0\in\mathcal A$ be an equilibrium. Then the following estimates hold: Indeed, only the left-hand side of the second inequality has not been proved yet. Its proof follows in a straightforward way from definitions and the spectral mapping theorem (see [45] for more details). Note that, in contrast to the finite-dimensional case, this inequality may be strict if $\Phi$ is infinite-dimensional, since we have only one-sided spectral mapping theorem for the essential spectrum. We now turn to examples. Example 6.29. We return to Example 2.10, which is the simplest example. We have already proved that the fractal dimension of the corresponding attractor $\mathcal A$ is finite in $\Phi=L^2(0,\pi)$ and satisfies the upper bound (6.28). We now apply the volume contraction method and Proposition 6.28 to get more explicit upper and lower bounds. First we note that the corresponding solution semigroup is $C^\infty$-smooth here (see, for example, [12]), so all smoothness assumptions are automatically satisfied, and we only need to get good upper and lower bounds for the Lyapunov dimension and the instability indices. The operator $\mathcal L(u(t))$ for the equation of variations now reads $\mathcal L(u)=\partial_x^2-3u^2+a$, and the eigenvalues of the operator $-\partial_x^2$ with the Dirichlet boundary conditions are $\lambda_n=n^2$. Therefore, by the min-max principle, for integer values of $d$ we have
$$
\begin{equation*}
\operatorname{Tr}_d(\mathcal L(u))\leqslant \operatorname{Tr}_d(\partial_x^2+a)= -\sum_{n=1}^d\lambda_n+ad=-\frac{d(d+1)(2d+1)}{6}+ad\geqslant -\frac{d^3}{3}-ad.
\end{equation*}
\notag
$$
We recall that $u_0=0$ is an equilibrium and the first inequality in the above formula is attained at this equilibrium. This gives us
$$
\begin{equation}
\dim_{\rm L}(S(t),\mathcal A)=\dim_{\rm L}(S(t),0)= \dim_{\rm L}(\mathcal L(0))=\dim_{\rm L}(\partial_x^2+a),
\end{equation}
\tag{6.49}
$$
where the second equality is due to the fact that the spectral mapping theorem holds for self-adjoint operators. Note that the right-hand side of equality (6.49) is easy to compute explicitly, so the Lyapunov dimension of the attractor $\mathcal A$ has not only been estimated, but can also be computed explicitly. We will not write out this formula, but only mention that, for large $a$ we have
$$
\begin{equation}
\dim_{\rm H}(\mathcal A,\Phi)\leqslant\dim_{\rm f}(\mathcal A,\Phi)\leqslant \dim_{\rm L}(S(t),\mathcal A)\sim \sqrt{3a}\,.
\end{equation}
\tag{6.50}
$$
Let us now look at lower bounds for Hausdorff and fractal dimensions of the attractor $\mathcal A$. We see that the instability index $\mathcal N_+(0)$ is of order $\sqrt{a}$ . Moreover, since the equation under consideration possesses a global Lyapunov function, the corresponding attractor is the union of the unstable sets of all equilibria, that is,
$$
\begin{equation*}
\dim_{\rm H}(S(t),\mathcal A)\sim\max_{u_0\in\mathcal R}\mathcal N_+(u_0)= \mathcal N_+(0)\sim \sqrt{a}\,.
\end{equation*}
\notag
$$
So the Hausdorff dimension of the attractor can also be explicitly computed. In contrast to this, for the fractal dimension we only have the two-sided inequality
$$
\begin{equation}
\sqrt{a}\sim \mathcal N_+(0)\sim\dim_{\rm H}(\mathcal A,\Phi)\leqslant \dim_{\rm f}(\mathcal A,\Phi)\leqslant\dim_{\rm L}(S(t),\mathcal A)= \dim_{\rm L}(\mathcal L(0))\sim\sqrt{3a}\,.
\end{equation}
\tag{6.51}
$$
Note that even in this simplest example the fractal dimension may a priori be larger than the Hausdorff one due to the non-trivial intersections of stable and unstable manifolds of equilibria, so we basically do not have anything more than the two-sided bounds (6.51). Remark 6.30. Note that the positivity of the Lyapunov dimension $\dim_{\rm L}(S(t),\mathcal A)$ does not imply here the presence of a chaotic dynamics of the system under consideration (in our example the system is gradient and the dynamics is trivial). This should not be misleading, since in contrast to the classical dynamics, where the Lyapunov exponents for individual trajectories (which exist for almost all trajectories with respect to the “physical measure”) are usually considered (see, for example, [116]), we consider here the so-called uniform Lyapunov exponents, which may be positive even for the trivial gradient dynamics. Note also that the coincidence of the Lyapunov dimension of the whole attractor with the dimension of an equilibrium is not limited to gradient systems. For instance, exactly the same happens (for some different reasons) in the case of the chaotic Lorenz attractor considered in Example 2.8. We notice also that, in the example considered above, the instability index $\mathcal N_+(u_0)$ and the Lyapunov dimension $\mathcal L(u_0)$ have the same asymptotic behaviour as $a\to\infty$ (up to the non-essential multiplier $\sqrt3$ ), which allowed us to describe the sharp asymptotic behaviour of $\dim_{\rm f}(\mathcal A,\Phi)$ as $a\to\infty$. This is somehow typical for parabolic equations but can be easily fail, for example, for hyperbolic equations. Example 6.31. In particular, this is not true for the equation considered in Example 6.26, to which we return now. Indeed, due to assumptions (6.40), the eigenvalues $\nu_F$ of the linearized operator (6.42) at any equilibrium $u_0$ are asymptotically very close to the eigenvalues $\nu_0$ of the operator which corresponds to $F=0$, namely, for all $N\in\mathbb N$ (see [220]). The eigenvalues which correspond to $F=0$ are easy to compute: Finally, if we embed such a construction in the attractor $\mathcal A$ of (6.39), then we end up with the desired sharp lower bound: Exactly this approach was realized in [220]. Namely, consider the following model decoupled system of ODEs:
$$
\begin{equation}
u_0''=u_0-u_0^3,\qquad u_n''+\gamma u_n'+\lambda_n u_n=0,\quad n=1,2,\dots,
\end{equation}
\tag{6.54}
$$
where $\gamma>0$ is small enough, and $\gamma^2-4\lambda_n>0$. Then this system has a zero equilibrium and a homoclinic loop $\{u_0(t),0\}$ to it. Moreover, $u_0(t)\sim e^{-|t|}$ as $t\to\pm\infty$, and the remaining eigenvalues (which correspond to equations for $u_n$, $n=1,2,\dots$) are
$$
\begin{equation*}
\mu_n^{\pm}=\frac{-\gamma\pm\sqrt{\gamma^2-4\lambda_n}}{2}\,.
\end{equation*}
\notag
$$
Thus, $\operatorname{Re}\mu_n^{\pm}=-\gamma/2$ and conditions (6.53) are satisfied for $\lambda=1$ if $\gamma$ is small enough. It was proved in [220] that for every $m\in\mathbb N$, every $\varepsilon>0$ and $\gamma>0$ small enough, there exists a perturbation $F_i$, $\|F_i\|_{C^m}\leqslant\varepsilon$, such that the perturbed system
$$
\begin{equation}
\begin{gathered} \, u_0''=u_0-u_0'+F_0(u_0,u_0',u,\partial_t u), \\ u_n''+\gamma u_n'+\lambda_nu_n=F_n(u_0,u_0',u_1,u_1',\dots),\qquad n=1,2,\dots, \end{gathered}
\end{equation}
\tag{6.55}
$$
possesses a $C\gamma^{-1}$-dimensional invariant torus in a small neighbourhood of the unperturbed homoclinic loop. Moreover, only finitely many ($n_0=C\gamma^{-1}$) modes are really perturbed and the perturbation depends on $u_n$, $n\leqslant n_0$, only. It remains to note that equations (6.55) can easily be embedded in a system of the form (6.39), where the $C^m$-norm of the nonlinearity $F$ is uniformly bounded as $\gamma\to0$. This shows that the upper bounds $\dim_{\rm f}(\mathcal A,E)\sim C\gamma^{-1}$ are indeed sharp. Remark 6.32. Note that, in general, we need a large number of parameters (of order proportional to $\dim_{\rm L}(S(t),u_0)$) in order to be able to bear the torus and use this scheme for constructing examples with large fractal dimension of the attractor. We expect that this method will be helpful for other types of equations of hyperbolic type, for example, for damped Euler equations and their various regularizations. 7. Inertial manifolds and finite-dimensional reductionIn the previous section we constructed a reduced system of ODEs (an inertial form of the initial PDE), which captures the limit dynamics on the attractor $\mathcal A$, relying on the finiteness of its fractal dimension and Mané’s projection theorem (see (6.3)). We also pointed out the main drawback of such a reduction, namely, a drastic loss of smoothness. In this section we discuss an alternative approach to the finite-dimensional reduction, which is based on the theory of centre manifolds (or more generally, on the theory of normally-hyperbolic invariant manifolds), which requires stronger assumptions on the dynamical system under consideration, but gives a much more appropriate construction of the inertial form, which is suitable for applications. To the best of our knowledge, this approach was proposed by Mané [168] and has become very popular after the paper of Foias, Sell, and Temam [76] (see also [2], [175], [195], [199], [243], and the references therein). Following this approach, the main object of the theory is not an attractor, but a globally stable finite-dimensional invariant submanifold of the initial phase space, which is typically normally hyperbolic and, for this reason, exponentially attractive. The reduced inertial form in this situation is nothing else than the restriction of the initial PDE to this invariant manifold and is typically as smooth as the manifold. Keeping in mind possible applications to hydrodynamics and the hope to understand turbulence, this manifold has been referred to as the inertial manifold in reference to the so-called inertial scale in the conventional theory of turbulence (see, for example, [81] and the references therein). We start with the formal definition of an inertial manifold. Definition 7.1. Let $S(t)\colon\Phi\to\Phi$ be a dynamical system acting in a Banach space $\Phi$. Then a strictly invariant finite-dimensional Lipschitz submanifold $\mathcal M$ of the phase space $\Phi$ is an inertial manifold for $S(t)$ if it possesses an exponential tracking property (an asymptotic phase) in the following form: for any semi-trajectory $u(t)=S(t)u_0$, $t\geqslant 0$, of the dynamical system under consideration there exists a semi-trajectory $\overline u(t)=S(t)\overline u_0$ with $\overline u_0\in\mathcal M$ such that where the positive constant $\alpha$ and a monotone function $Q$ are independent of $u_0$ and $t$. Remark 7.2. In applications an inertial manifold is often constructed as the graph of a Lipschitz function. Namely, assume that $\Phi=\Phi_+\oplus\Phi_-$ is represented as a direct sum of two Banach spaces (the corresponding projectors are denoted by $\Pi_+$ and $\Pi_-$, respectively) where $\dim\Phi_+=N$. Then $u(t)=\Pi_+u(t)+\Pi_-u(t)=u_+(t)+u_-(t)$. The variables $u_+(t)$ and $u_-(t)$ are treated as the “slow” and “fast” ones, respectively, and the manifold $\mathcal M$ slaves the fast variables $u_-(t)$ to the slow ones $u_+(t)$, that is, $u_-(t)=M(u_+(t))$ for some at least Lipschitz function $M\colon\Phi_+\to\Phi_-$, so that $\mathcal M$ is treated as the graph of this function: We note from the very beginning that in applications one usually constructs an inertial manifold not for the original PDE, but for a properly modified one (the so-called “prepared” equation), whose solutions have the same asymptotic behaviour, but it itself has a better structure, for example, its nonlinearity is globally Lipschitz continuous. Since an inertial manifold is a kind of a (global) centre manifold, it is not unique in general and we need to cut-off the nonlinearity properly to restore the uniqueness and the possibility to find it using Banach’s contraction theorem. In contrast to ODEs, this preparation procedure may be very delicate: sometimes it is enough just to cut-off the nonlinearity outside an appropriate absorbing set, thus making it globally Lipschitz (this usually works when the spectral gap conditions are satisfied): see [76]; but in other cases, you may need to embed your initial PDEs in a larger system of PDEs, do some diffeomorphisms to restore the spectral gap conditions, change the leading-order differential operator, and so on (see [133], [126], [131], [128], [166]). In addition, if you want to improve the smoothness of an inertial manifold, you need to change the vector field at points of the attractor as well (in an accurate way in order not to affect the dynamics on the attractor, but kill the resonances); see [136]. We do not go into further details here and will always assume in what follows that we have already made some of these “preparations” and the nonlinearity is already globally Lipschitz. 7.1. Spectral gap conditions and inertial manifoldsWe start our exposition with the classical theory of inertial manifolds, which usually deals with a semilinear equation of the form:
$$
\begin{equation}
\partial_t u+Au=F(u), \quad u\big|_{t=0}=u_0\in\Phi.
\end{equation}
\tag{7.3}
$$
We assume for simplicity that $\Phi$ is a real Hilbert space, $A\colon D(A)\to\Phi$ is a positive self-adjoint linear operator with compact inverse, and $F\colon\Phi\to\Phi$ is a given nonlinearity, which is Lipschitz continuous with Lipschitz constant $L$:
$$
\begin{equation}
\|F(u_1)-F(u_2)\|_{\Phi}\leqslant L\|u_1-u_2\|_\Phi,\qquad u_1,u_2\in\Phi.
\end{equation}
\tag{7.4}
$$
From the Hilbert–Schmidt theorem we know that the operator $A$ possesses a complete basis of eigenvectors $\{e_i\}_{i=1}^\infty$ with the corresponding eigenvalues $\{\lambda_i\}_{i=1}^\infty$, which we enumerate in the non-decreasing order. The key result of the theory is the following theorem. Theorem 7.3. Let the nonlinearity $F$ satisfy (7.4), and let $N\in\mathbb N$ be such that the following spectral gap condition is satisfied: Then equation (7.3) possesses an $N$-dimensional inertial manifold which can be represented as the graph of a Lipschitz-continuous function $M\colon\Phi_+\to\Phi_-$, where $\Phi_+$ is spanned by the first $N$ eigenvectors of $A$ and $\Phi_-$ is the orthogonal complement of $\Phi_+$. Moreover, if $F\in C^{1+\varepsilon}(\Phi,\Phi)$ for a sufficiently small $\varepsilon>0$, then the corresponding map $M$ is also $C^{1+\varepsilon}$-smooth. We also mention that the attraction exponent $\alpha$ in the definition of an inertial manifold can be chosen as $\alpha=(\lambda_N+\lambda_{N+1})/2$. Idea of the proof. The construction of the manifold is based on perturbation arguments where the nonlinearity $F$ is considered as a perturbation. Indeed, for $F\equiv0$ we have $\mathcal M=\Phi_+$, all trajectories on this manifold grow not faster than $e^{-\theta t}$ as $t\to-\infty$, where $\theta\in(\lambda_N,\lambda_{N+1})$, and this is a determining property for the manifold $\mathcal M$.
This observation hints at the way to construct the desired inertial manifold $\mathcal M$, namely, we need to solve the problem
$$
\begin{equation}
\partial_t u+Au=F(u),\quad t\leqslant0,\qquad \Pi_+u\big|_{t=0}=u_+\in\Phi_+
\end{equation}
\tag{7.6}
$$
backward in time in the weighted space $L_{e^{\theta t}}(\mathbb{R}_-,\Phi)$, where $\theta\in(\lambda_N,\lambda_{N+1})$ is chosen in an optimal way ($\theta=(\lambda_{N}+\lambda_{N+1})/2$ in our case). Then the desired map $M$ will be found as follows: $M\colon u_+\to\Pi_-u(0)$ (see [243] for more details). Note that the idea to use here the space $L^2$ with respect to time belongs to Miklavcic [175]. The use of the space $C_{e^{\theta t}}(\mathbb{R}_-,\Phi)$, which appears more natural at first glance, leads to the extra multiplier $\sqrt{2}$ on the right-hand side of (7.5).
Following [175], [243] (see also [76]), we solve equation (7.6) using Banach’s contraction theorem, and the most important step here is to find the norm of the solution operator $\mathcal L_\theta\colon h\to v$ for the following linear problem on the whole line $t\in\mathbb{R}$:
$$
\begin{equation}
\partial_t v+Av=h(t),\qquad h\in L^2_{e^{\theta t}}(\mathbb{R},\Phi),
\end{equation}
\tag{7.7}
$$
in the weighted space $L^2_{e^{\theta t}}(\mathbb{R},H)$, namely, to verify that
$$
\begin{equation}
\|\mathcal L_\theta\|_{\mathcal L(L^2_{e^{\theta t}},L^2_{e^{\theta t}})}= \max\biggl\{\frac{1}{\theta-\lambda_N}\,, \frac{1}{\lambda_{N+1}-\theta}\biggr\}.
\end{equation}
\tag{7.8}
$$
The proof of this estimate is very elementary since the problem is reduced to the analogous one for the scalar ODEs which are the equations on the Fourier amplitudes for (7.7). An elementary calculation shows that
$$
\begin{equation}
\|\mathcal L_{\theta,n}\|_{\mathcal L(L^2_{e^{\theta t}},L^2_{e^{\theta t}})} =\frac1{|\lambda_n-\theta|}
\end{equation}
\tag{7.9}
$$
and Parseval’s equality gives us the relation $\|\mathcal L_\theta\|=\max_{n\in\mathbb N}\|\mathcal L_{\theta,n}\|$ (here we see the advantage of using weighted $L^2$-spaces). This yields the desired estimate (7.8).
The rest of the proof is also straightforward. We just invert the linear part of (7.6) (after an appropriate restriction to the negative semi-axis $t\leqslant0$) and apply Banach’s contraction theorem. The optimal exponent $\theta=(\lambda_N+\lambda_{N+1})/2$ gives us the value $2/(\lambda_{N+1}-\lambda_N)$ for the norm of $\mathcal L_\theta$, so the Lipschitz constant of the composition $\mathcal L_\theta$ with $F$ does not exceed $2L/(\lambda_{N+1}-\lambda_N)<1$ due to the spectral gap condition. Thus the map constructed is indeed a contraction. The exponential tracking property is also an almost immediate corollary of (7.8) and Banach’s contraction theorem (see [243] for the missed details). Remark 7.4. The spectral gap condition (7.5) can be generalized to the case where the operator consumes smoothness. Namely, instead of (7.4) assume that the operator $F$ is globally Lipschitz as a map from $\Phi$ to $\Phi^{-s}:=D(A^{-s/2})$, for some $s\in(0,2)$, with the same Lipschitz constant $L$. Then an analogue of (7.5) reads We mention here the paper [33] (see also [31] and [182] for weaker results), where the hyperbolic relaxation
$$
\begin{equation*}
\varepsilon\partial_t^2 u+\partial_t u+Au=F(u),\qquad \varepsilon>0,
\end{equation*}
\notag
$$
of (7.3) is considered. It was shown there that the sharp spectral gap condition for this problem coincides with (7.5) (where $\lambda_n$ are the eigenvalues of $A$) and is independent of $\varepsilon$ for $\varepsilon$ small enough. Inertial manifolds for elliptic boundary problems in cylindrical domains were studied in [7] and [172]. We also mention that the necessary conditions for the existence of inertial manifolds are often formulated in the spirit of the theory of dynamical systems in terms of invariant cones. We do not present the details here and refer the interested reader to [2], [195], [243] for more details. Let us discuss some examples where the classical theory is applicable (more examples can be found in the survey [243], see also [215] and the references therein). We start with the system of reaction-diffusion equations in a bounded domain $\Omega$ of $\mathbb{R}^d$:
$$
\begin{equation}
\partial_t u=a\Delta_x u-f(u), \qquad u=(u^1,\dots,u^m),
\end{equation}
\tag{7.12}
$$
endowed with appropriate boundary conditions. We assume that the diffusion matrix $a$ is self-adjoint and positive definite and $f$ is globally Lipschitz. Then the operator $A:=-a\Delta_x$ is self-adjoint and positive (non-negative in the case of periodic or Neumann boundary conditions) in $\Phi=[L^2(\Omega)]^m$ and, due to Weyl’s theorem, we have so the validity of spectral gap conditions depends strongly on the dimension $d$. When $d=1$, we have infinitely many values of $N$ such that $\lambda_{N+1}-\lambda_N\geqslant cN$ for some positive $c$, so we have spectral gaps of any size, and therefore, for any Lipschitz constant $L$ of $f$, we may find $N$ satisfying the spectral gap condition and this guarantees the existence of an inertial manifold. The case $d=2$ is more interesting. In this case Weyl’s theorem can guarantee only that $\lambda_{N+1}-\lambda_N\geqslant c$ for infinitely many values of $N$, so we may apply Theorem 7.3 only if $L$ is small enough. However, we may still have spectral gaps of any size in the spectrum of the Laplacian despite the relation $\lambda_n\sim Cn$. For instance, for $\Omega=[-\pi,\pi]^2$ with periodic boundary conditions, we have infinitely many values of $N$ satisfying (see [190]), so an inertial manifold exists in the case of periodic boundary conditions. The ideal situation from the point of view of inertial manifolds is the case of the Laplace–Beltrami operator on the $d$-dimensional sphere $\mathbb S^d$, where the inequality $\lambda_{N+1}-\lambda_N\geqslant c\lambda_N^{1/2}$ has infinitely many solutions in any space dimension $d$. Thus, reaction-diffusion equations of the form (7.12) on a sphere always have inertial manifolds in any space dimension. Note that, to the best of our knowledge, the problem of the existence of spectral gaps of any size for the 2D Laplacian in a bounded domain $\Omega\subset\mathbb{R}^2$ is completely open. On the one hand, we do not know any examples where such gaps do not exist and, on the other hand, we do not know any reasonably general classes of domains that possess this property.In contrast to this, in dimension three or higher spectral gaps of an arbitrary size exist in very exceptional cases only (like sphere), so the classical inertial manifolds theory is not very helpful for 3D reaction-diffusion equations. Nevertheless, spectral gap conditions are still satisfied for higher-order equations like the Swift–Hohenberg equation since for the bi-Laplacian Weyl’s theorem gives $\lambda_n\sim c n^{4/3}$, and this guarantees the existence of spectral gaps of any size.Now consider equations with nonlinearities decreasing the regularity. The classical example here is the Kuramoto–Sivashinski equation in 1D:
$$
\begin{equation*}
\partial_t u+\partial_x^4 u-a\partial_x^2 u+\partial_x(u^2)=0,\quad u\big|_{t=0}=u_0, \quad a>0,
\end{equation*}
\notag
$$
in $\Omega=[-\pi,\pi]$ endowed with periodic boundary conditions. Since we have the conservation law here: $\langle u\rangle:=\dfrac{1}{2\pi}\displaystyle\int_{-\pi}^\pi u(x)\,dx$, we must consider this equation in the phase space $\Phi:=L^2(-\pi,\pi)\cap\{\langle u\rangle =0\}$. It is well known (see [89], [215]) that this equation generates a dissipative semigroup $S(t)$ in $\Phi$ which possesses a smooth absorbing set. We set $A:=\biggl(\partial_x^2-\dfrac{a}{2}\biggr)^2+1$ and $F(u)=\dfrac{a^2}{4u}+u-\partial_x(u^2)$. After an appropriate cut-off, the nonlinearity $F$ becomes globally Lipschitz as a map from $H$ to $H^{-1}(\Omega)=D(A^{1/4})$, an therefore we need to check the spectral gap condition (7.10) for $s=1/2$. We know that $\lambda_n\sim c n^4$, and therefore
$$
\begin{equation*}
\frac{\lambda_{N+1}-\lambda_N}{\lambda_{N}^{1/4}+\lambda_{N+1}^{1/4}} \sim c_1N^2
\end{equation*}
\notag
$$
and we have spectral gaps of arbitrarily large size. Thus, this equation possesses an inertial manifold. One more example, which is interesting for what follows, is given by an 1D system of reaction-diffusion advection equations:
$$
\begin{equation}
\partial_t u-\partial_x^2 u+u=f(u,\partial_x u), \quad u=(u^1,\dots,u^m),\quad u\big|_{t=0}=u_0,
\end{equation}
\tag{7.14}
$$
endowed with Dirichlet, Neumann, or periodic boundary conditions. We assume that $f$ is smooth and both $f'_u$ and $f'_{\partial_x u}$ are uniformly bounded. Here $A=-\partial_x^2+1$ and the nonlinearity $f$ consumes one unit of smoothness, so $s=1$. Since $\lambda_n\sim c n^2$, the corresponding spectral gap condition reads
$$
\begin{equation*}
\frac{\lambda_{n+1}-\lambda_n}{\lambda_{n}^{1/2}+ \lambda_{n+1}^{1/2}}\sim c>L.
\end{equation*}
\notag
$$
Thus, we have an inertial manifold via the classical theory if the Lipschitz constant $L$ of $f$ is small enough. A slightly more accurate analysis shows (see [132], [168]) that the size of $f'_u$ is not essential and only $L_1:=\sup_{u,v\in\mathbb{R}}|f'_v(u,v)|$ should be small. As we will see below, this observation is crucial for the recent theory of inertial manifolds for these equations, which is discussed in the next subsection. We now turn to the smoothness of inertial manifolds. First we note that, in the case where $\mathcal M$ is the graph of a function $M\colon\Phi_+\to\Phi_-$ and $\Phi_+$ is a spectral subspace of the operator $A$, the inertial form (6.3) for equation (7.3) is significantly simplified:
$$
\begin{equation}
\partial_t u_++Au_+=\Pi_+F(u_++M(u_+)),\qquad u_+\in\Phi_+\cong\mathbb{R}^N,
\end{equation}
\tag{7.15}
$$
and we see that the regularity of the reduced equations is determined by the smoothness of the map $M$. Theorem 7.3 guarantees that these reduced equations are $C^{1+\varepsilon}$-smooth for some small $\varepsilon>0$ if the spectral gap conditions are satisfied and $F$ is smooth. This regularity cannot be improved in general since, similarly to the theory of centre manifolds (or, more generally, the theory of normally hyperbolic invariant manifolds), there are obstacles to a further regularity of $\mathcal M$ (see, for instance, a counterexample of Sell [40] related to resonances). The nature of these obstacles can be explained as follows. Let us formally differentiate equation (7.6) in order to get an equation for the Fréchet derivative of the map $M$:
$$
\begin{equation}
\partial_t v+Av=F'(u(t))v,\quad \Pi_+v\big|_{t=0}=\xi\in\Phi_+,
\end{equation}
\tag{7.16}
$$
where $M'(u_+)\xi:=\Pi_-v(0)$. Since $\|F'(u(t))\|\leqslant L$, the spectral gap conditions still allow us to solve this equation uniquely in the space $L^2_{e^{\theta t}}(\mathbb{R}_-,\Phi)$ and define the map $M'(u_+)$. A slightly more accurate analysis shows that the map obtained is Hölder continuous with small positive Hölder exponent $\varepsilon>0$ and the function $M$ is $C^{1+\varepsilon}$ (see, for example, [136], [243]). The situation changes drastically when we differentiate equation (7.16) once more and look at the second derivative:
$$
\begin{equation}
\partial_t w+Aw-F'(u(t))w=F''(u(t))[v_\xi(t),v_\eta(t)]=: h_{\xi,\eta}(t),\quad \Pi_+w\big|_{t=0}=0,
\end{equation}
\tag{7.17}
$$
where $v_\xi$ and $v_\eta$ are the solutions of (7.16) with the initial data $\xi$ and $\eta$, respectively. The problem here is that $v_\xi,v_\eta\in L^2_{e^{\theta t}}(\mathbb{R}_-,\Phi)$ for some $\theta$, (in any case, we can guarantee the existence of a solution of (7.16) on the basis of Banach’s contraction theorem only for such values of $\theta$). Therefore, $h_{\xi,\eta}\in L^2_{e^{2\theta}}(\mathbb{R}_-,\Phi)$ (since we have the parabolic smoothing property, the product of two solutions belonging to a weighted $L^2$ is also in $L^2$ with an appropriate weight). But in order to solve (7.17), we need the exponent $2\theta$ to satisfy (7.18), and this is possible only if Analogously, if we want the inertial manifold to be $C^s$-smooth for some $s>1$, then we need the following spectral gap: This condition is actually sharp, and there is an example due to Sell [40] of an equation of the form (7.3) which possesses a $C^{2-\varepsilon}$-smooth inertial manifold, but does not possess any $C^2$-smooth inertial manifold (see also [136] and the references therein). Note that there is a principal difference between conditions (7.5) and (7.19), namely, the first condition requires the existence of gaps of arbitrarily large size in the spectrum of $A$ and can be satisfied at least for some elliptic operators in bounded domains (for example, in the case of low space dimension). In contrast to this, condition (7.19) requires exponentially big lacunas in the spectrum (for example, $\lambda_n=a^n$ with $a>2$), which is difficult to expect in the case of elliptic operators. In fact, we do not know any examples of such operators $A$ for which inequality (7.19) is solvable with respect to $N$ for any Lipschitz constant $L$. Thus, inertial manifolds constructed using Theorem 7.3 are never $C^2$-smooth in general if more or less realistic applications are considered. The only exception is the case of local bifurcations and associated local centre manifolds, where $\lambda_N\sim 0$ and $L$ can be chosen arbitrarily small by decreasing the size of the neighbourhood. Then condition (7.20) allows us to construct invariant manifolds of any finite smoothness. Nevertheless, there is a possibility to overcome (at least partially) the smoothness problem for inertial manifolds by increasing the dimension of the manifold and cutting off the nonlinearity $F$, namely, the following result was proved in [136]. Theorem 7.5. Let the assumptions of Theorem 7.3 hold. Assume, in addition, that $F\in C^\infty(\Phi,\Phi)$ and the following stronger version of the spectral gap conditions holds: 1) the manifold $\mathcal M_1$ remains an inertial manifold for the modified problem 2) equation (7.22) possesses a $C^m$-smooth inertial manifold $\mathcal M_m$ such that $\mathcal M_1$ is a normally hyperbolic exponentially stable invariant submanifold of $\mathcal M_m$; 3) the nonlinearity $F_m$ is $\delta$-close to $F$ in the $C^1$-norm. Idea of the proof. Thanks to (7.21), we have infinitely many spectral gaps suitable for constructing inertial manifolds. The first of them is used to construct the manifold $\mathcal M_1$. The key idea is to use the second spectral gap to solve equation (7.17) for the second derivative; then the third gap is used to find the third derivative, and so on. Then, for every $u\in\mathcal M_1$, the above procedure gives us an $m$-jet which should correspond to the $m$-smooth extension $\mathcal M_m$ of the manifold $\mathcal M_1$. Such a jet can be constructed in many ways, and the most difficult part of the proof is to fix it so that the compatibility conditions in Whitney’s extension theorem are satisfied. Then we get the desired manifold $\mathcal M_m$ by this theorem, and after that it is already not difficult to define the correction $F_m$ of the initial nonlinearity $F$ in such a way that $\mathcal M_m$ is an invariant manifold for the modified system (7.22); see [136] for the details. Remark 7.6. Note that the construction in Theorem 7.5 replaces the requirement (7.20) that one “huge” spectral gap exists (which is almost never satisfied) by the existence of many relatively small gaps, which is satisfied if we are able to satisfy (7.5) for any Lipschitz constant $L$ (so, this is not a big extra restriction). Note also that this theorem allows us to interpret the $C^{1+\varepsilon}$-smooth inertial form (7.15) (which is a system of ODEs in $\mathbb{R}^{N_1}$) as the reduced equations on the normally hyperbolic invariant manifold for the extended $C^m$-smooth system of ODEs in $\mathbb{R}^{N_m}$, which is the inertial form related to the inertial manifold $\mathcal M_m$. Of course, such an extension does not exist for a general non-smooth system of ODEs and is strongly related to the fact that the system of ODEs under consideration is obtained as a reduction of the smooth infinite-dimensional system (7.3). 7.2. Beyond the spectral gap conditionsWe now turn to the case where the spectral gap condition (7.5) is violated. We start our exposition with the sharpness of the assumptions of Theorem 7.3. Assume that (7.5) is not satisfied for some fixed $N\in\mathbb N$, namely, that $\lambda_{N+1}-\lambda_N<2L$. Consider the following linear version of system (7.3):
$$
\begin{equation}
\begin{gathered} \, \frac{d}{dt} u_n+\lambda_n u_n=0,\qquad n\ne N,N+1, \\ \frac{d}{dt}u_N+\lambda_Nu_N=Lu_{N+1},\qquad \frac{d}{dt}u_{N+1}+\lambda_{N+1}u_{N+1}=-Lu_N. \end{gathered}
\end{equation}
\tag{7.23}
$$
Then for the corresponding $F(u)=Fu$ we have $\|F\|_{\mathcal L(\Phi,\Phi)}=L$, so $L$ is indeed a Lipschitz constant for the perturbation $F(u)$. Then the eigenvalues of the perturbed system associated with the invariant subspace spanned by $e_N$ and $e_{N+1}$,
$$
\begin{equation*}
\mu_N^\pm=\frac{\lambda_{N+1}+\lambda_N}{2} \pm\sqrt{\biggl(\frac{\lambda_{N+1}-\lambda_N}2\biggr)^2-L^2}\,,
\end{equation*}
\notag
$$
are complex conjugate with non-zero imaginary part. For this reason we cannot decouple $u_N(t)$ and $u_{N+1}(t)$, and an invariant $N$-dimensional manifold (linear subspace) with the base $\Phi_+$ cannot exist. The same example shows that we also do not have a normally hyperbolic invariant subspace of dimension $N$ if the spectral gap condition is violated for $N$ (see, for instance, [197], [243] for more details). Now assume that (7.5) is violated for all $N\in\mathbb N$, that is, Then it is natural to find examples where we have no inertial manifold for any dimension $N$. Such examples are often based on the following lemma (see [197], [243]) and its generalizations.Lemma 7.7. Assume that the nonlinearity $F$ in equation (7.3) belongs to $C^1$, and let $u_\pm\in\Phi$ be two equilibria of this equation. Let $\mathcal L_{u_\pm}:=-A+F'(u_{\pm})$ be the linearization of (7.3) at $u=u_{\pm}$. Assume that the spectrum $\sigma(\mathcal L_{u_+})$ consists of complex conjugate eigenvalues with non-zero imaginary parts and the spectrum of $\sigma(\mathcal L_{u_-})$ contains one simple positive real eigenvalue and the rest of it consists of complex conjugate eigenvalues with non-zero imaginary parts. Then problem (7.3) does not possess any finite-dimensional $C^1$-smooth inertial manifold. Sketch of the proof. Indeed, assume that such a manifold $\mathcal M$ exists. Consider its tangent planes $\mathcal T_{u_\pm}\mathcal M$ at $u=u_{\pm}$. Since the manifold is invariant, the planes $\mathcal T_{u_{\pm}}\mathcal M$ are invariant with respect to $\mathcal L_{u_\pm}$. Therefore, Since the equation is real-valued and the spectrum of $\mathcal L_{u_+}$ does not contain real eigenvalues, we conclude that $\dim\mathcal M=\dim\mathcal T_{u_+}\mathcal M$ is even. On the other hand, since the inertial manifold always contains an attractor and the attractor always contains an unstable manifold, the direction of the simple real eigenvector of $\mathcal L_{u_-}$ must lie in $\mathcal T_{u_-}\mathcal M$. Then the analogous arguments show that $\dim\mathcal M=\dim\mathcal T_{u_-}\mathcal M$ is odd. This contradiction proves that the inertial manifold $\mathcal M$ cannot exist. Corollary 7.8. Let $\lambda_1<L$, and let assumption (7.24) be satisfied. Then there exists a smooth bounded and globally Lipschitz nonlinearity $F$ with Lipschitz constant $L$ such that equation (7.3) does not possess any $C^1$-smooth finite-dimensional inertial manifold. Sketch of the proof. According to the previous lemma, we only need to construct $F$ in equation (7.3) so that it possesses two equilibria $u=u_\pm$ such that the linearizations near $u_+$ and $u_-$ have the form
$$
\begin{equation}
\frac{d}{dt}v_{2n-1}+\lambda_{2n-1}v_{2n-1}=Lu_{2n},\quad \frac{d}{dt}v_{2n}+\lambda_{2n}v_{2n}=-Lu_{2n-1},\quad n\in\mathbb N,
\end{equation}
\tag{7.25}
$$
and
$$
\begin{equation}
\begin{gathered} \, \frac{d}{dt}v_{1}+\lambda_{1}v_{1}=Lu_{1}, \\ \frac{d}{dt}v_{2n}+\lambda_{2n}v_{2n}=Lu_{2n+1},\quad \frac{d}{dt}v_{2n+1}+\lambda_{2n+1}v_{2n+1}=-Lu_{2n},\quad n\in\mathbb N \end{gathered}
\end{equation}
\tag{7.26}
$$
respectively. Then all assumptions of the previous corollary will be satisfied. The construction of such a nonlinearity $F$ is straightforward and we drop it here (see [243] for the details). Remark 7.9. Note that the fractal dimension of the global attractor $\mathcal A$ which corresponds to equation (7.3) is always finite if $F$ is bounded and globally Lipschitz. Thus, according to the common paradigm, the corresponding dynamics on the attractor should be “finite-dimensional”. The result of Corollary 7.8 does not essentially contradict this heuristic principle: it just tells us that the dimension of the inertial “manifold” should be different in different parts of the phase space. For instance, instead of an inertial manifold, one may try to consider an “inertial CW-complex” with finite-dimensional dynamics on it. As we will see below, this actually also does not work and the reduced dynamics may be essentially infinite-dimensional despite the finiteness of the fractal dimension. The further progress in understanding the obstacles to the existence of inertial manifolds and their consequences is related to the recently discovered connections of the theory of inertial manifolds with Floquet’s theory for infinite-dimensional differential equations (see [64], [133], [139], [243], and the references therein). Namely, consider the following linear time-periodic parabolic equation in a Hilbert space $\Phi$: where $L(t)$ is a linear operator $T$-periodic in time and satisfying $\|L(t)\|_{\mathcal L(\Phi,\Phi)}\leqslant L$. Roughly speaking, we fix the operator $L(t)$ in such a way that (7.27) is close to (7.25) and (7.26) on the first and second half-periods, respectively. Moreover, this can be done in such a way that the period map $\mathcal P\colon u_0\to u(t)$ has the form of an infinite Jordan cell:
$$
\begin{equation}
\begin{aligned} \, \ldots,\mathcal Pe_{2n}&=\mu_{2n}e_{2n-2},\quad \dots,\quad \mathcal Pe_2=\mu_0e_1, \\ \mathcal Pe_1&=\mu_1e_3,\quad\dots,\quad\mathcal Pe_{2n+1}= \mu_{2n+1}e_{2n+3},\quad\dots, \end{aligned}
\end{equation}
\tag{7.28}
$$
where $\mu_n\sim e^{-\alpha |n|}$ for some positive $\alpha$. In particular, the map $\mathcal P$ is compact and $\sigma(\mathcal P)=\{0\}$, so the corresponding equation (7.27) does not possess any Floquet multipliers and any solution of it decays super-exponentially to zero:
$$
\begin{equation}
\|v(t)\|_{\Phi}\leqslant C\|v_0\|_\Phi e^{-\alpha t^2/2}
\end{equation}
\tag{7.29}
$$
(see [64], [243] for more details). The next step is to realize the operator $L(t)$ in the form of $L(u_0(t))$, where $u_0(t)$ is a time-periodic solution of an ODE which is interpreted as an equation for the “zero-mode” of the abstract equation (7.3). This will give us a super-exponentially attracting periodic orbit $u(t):=\{u_0(t),0\}$ in the system of the form (7.3), which belongs to the attractor. It only remains to modify the equations properly away from this periodic orbit in order to find another periodic orbit $\overline u(t)$ belonging to the attractor and converging super-exponentially fast to $u(t)$:
$$
\begin{equation}
\|u(t)-\overline u(t)\|_\Phi\leqslant Ce^{-\alpha t^2/2},\qquad u,\overline u\in\mathcal A
\end{equation}
\tag{7.30}
$$
(see [64], [243] for more details). The presence of these two trajectories on the attractor clearly excludes the existence of Lipschitz continuous inertial manifolds of any finite dimension. Moreover, an appropriate modification of this construction also excludes the possibility to embed the attractor $\mathcal A$ in any finite-dimensional Lipschitz or even $\log$-Lipschitz continuous (not necessarily invariant) submanifolds. Namely, the following result was proved in [64]. Theorem 7.10. Let the assumptions of Corollary 7.8 be satisfied. Then there exists a smooth and bounded nonlinearity $F$ with global Lipschitz constant $L$ such that there exist two solutions $u(t)$ and $\overline u(t)$ of equation (7.3) belonging to the attractor and satisfying (7.30). Moreover, the corresponding attractor cannot be embedded in any finite-dimensional $\log$-Lipschitz submanifold of $\Phi$. In particular, no inertial manifold exists for this equation. Remark 7.11. Since under the assumptions of Theorem 7.10 the fractal dimension of the attractor $\mathcal A$ is finite, thus due to Mané’s projection theorem we have a Hölder-continuous inertial form on the attractor, as well as its embedding in a Hölder-continuous submanifold of $\Phi$. Moreover, the Hölder exponent can be made as close to $1$ as we want by increasing the dimension of the manifold (see [193] and the references therein). Nevertheless, the super-exponential attraction of trajectories is not observed in the classical dynamics generated by smooth ODEs and can hardly be interpreted as a finite-dimensional phenomenon. Thus, we see some kind of infinite-dimensional limit dynamics on an attractor of finite fractal dimension. This phenomenon is not properly understood yet, and it definitely deserves further investigation. Remark 7.12. Note that it is relatively easy to construct counterexamples to Floquet’s theory on the level of abstract linear parabolic equations of the form (7.27), for instance, by constructing the desired operator $L(t)$ in the Fourier base of the operator $A$. The situation becomes much more complicated if we want (7.27) to be a true parabolic PDE or system of parabolic PDEs. The existence of counterexamples to Floquet’s theory on the level of parabolic PDEs in bounded domains was been an open problem till recently. This problem was affirmatively solved in [133], where the smooth space-time periodic $m\times m$ matrices $a(t,x)$ and $b(t,x)$ were constructed for $m\geqslant 4$ in such a way that all solutions of the corresponding linear reaction-diffusion-advection problem We now turn to examples where the inertial manifold still exists despite the fact that the spectral gap condition is not satisfied. We have seen above that these conditions are sharp on the level of abstract parabolic equations; however, there is still a possibility to relax them if some specific subclass of such equations is considered. For instance, the first such example was due to Sell and Mallet-Paret [166] where the inertial manifold for a scalar semilinear heat equation on the 3D torus $\Omega=[-\pi,\pi]^3$ was constructed. We return to this example below, but prefer to consider first an alternative method related to finding an appropriate transformations or/and embedding of the initial system into a larger one for which the spectral gap conditions are satisfied. We illustrate this approach by an example of the reaction-diffusion-advection system (7.32) endowed with Dirichlet boundary conditions. Namely, let us make the change of the independent variable $u=a(t,x)w$, where the matrix $a(t,x)$ will be specified in what follows. Then we arrive at the transformed equation
$$
\begin{equation}
\begin{aligned} \, \notag \partial_t w-\partial_x^2w&=\{a^{-1}(2\,\partial_xa+g(aw)a)\,\partial_xw\} \\ \notag &\qquad+\{a^{-1}(\partial_x^2a-\partial_t a+g(aw)\,\partial_xa)w+a^{-1}f(aw)\} \\ &=:\mathcal F_1(w)+\mathcal F_2(w). \end{aligned}
\end{equation}
\tag{7.33}
$$
We see that the operator $\mathcal F_2$ does not depend explicitly on $\partial_xw$ and this dependence is presented in $\mathcal F_1$ only, so the naive idea would be to kill the term $\mathcal F_1$ by an appropriate choice of the matrix $a=a(u)$, for instance, by fixing it as a solution of the matrix ODE
$$
\begin{equation*}
\frac{1}{2}\,\frac{d}{dx}a+g(u)a=0,\quad a\big|_{x=-\pi}=\operatorname{Id}.
\end{equation*}
\notag
$$
However, this naive idea will not work since the terms $\partial_t a$ and $\partial_x^2a$ will implicitly depend on $\partial_xw$ and the operator $\mathcal F_2$ will be not bounded from $\Phi:=H^1_0(-\pi,\pi)$ to $\Phi$. Fortunately, as we already mentioned in discussing equation (7.14), we need not kill the operator $\mathcal F_1$ completely: it is sufficient to make its Lipschitz constant (as a map from $\Phi$ to $L^2(-\pi,\pi)$) small enough and this hints at the suitable transformation: we fix $a=a(u)$ as a solution of the following problem:
$$
\begin{equation}
\frac{1}{2}\,\frac{d}{dt}a+g(P_Ku)a=0,\quad a\big|_{x=-\pi}=\operatorname{Id},
\end{equation}
\tag{7.34}
$$
where $P_K$ is some smoothing operator in $x$, for instance, we can fix it as the spectral projector onto the first $K$ eigenvectors of $-\partial_x^2$ in $\Omega=(-\pi,\pi)$ with Dirichlet boundary conditions and can set $K$ to be large enough. It was shown in [132] that $u=a(aw)w$ thus defined is indeed a diffeomorphism of the phase space $\Phi$, the Lipschitz norm of $\mathcal F_1\colon\Phi\to L^2(-\pi,\pi)$ can be made arbitrarily small by choosing $K$ large enough, and the map $\mathcal F_2\colon\Phi\to\Phi$ is globally Lipschitz continuous. Thus, an appropriate version of the spectral gap conditions is satisfied, and we have the following result (see [132] for the details). Theorem 7.13. Let the functions $f$ and $g$ be smooth and have finite supports. Then equation (7.32) endowed with Dirichlet boundary conditions possesses an inertial manifold. Remark 7.14. As we see from Remark 7.12 and Theorem 7.13, the existence or non-existence of an inertial manifold for the reaction-diffusion-advection problems depends strongly on the type of boundary conditions. This is related to the fact that the transformation $u=a(t,x)w$ we use preserves Dirichlet boundary conditions in a natural way, but does not preserve Neumann or periodic conditions. In the case of Neumann boundary conditions, there is a nice trick which allows us to overcome this problem, namely, we use that $v=\partial_xu$ satisfies Dirichlet boundary conditions and embed system (7.32) into the larger system We also mention that the general case of equation (7.14) can also be treated in a similar way by differentiating the equation with respect to time and embedding the problem into a larger system of equations of the form (7.32) (see [133] for more details). Thus, the assumption of smallness of the derivative $\partial_{u_x}f$ can be removed in the case of Dirichlet or Neumann boundary conditions, but cannot be relaxed in general for systems with periodic boundary conditions. Example 7.15. Consider a particular example of the forced Burgers equation: We note however that equation (7.35) is “too simple” to have any non-trivial dynamics, and its attractor always consists of a single equilibrium:
$$
\begin{equation}
\mathcal A:=\{G\},\quad \nu G''=(G^2)'+g,\quad G(\pm\pi)=0.
\end{equation}
\tag{7.36}
$$
Moreover, this equation does not generate any dynamical instability, and every trajectory of it is Lyapunov stable and even asymptotically stable. Indeed, let $u_1(t)$ and $u_2(t)$ be two solutions of (7.35), and let $v(t):=u_1(t)-u_2(t)$. Then $v$ solves
$$
\begin{equation*}
\partial_t v-\partial_{x}^2 v=\partial_x((u_1+u_2)v), \quad v\big|_{t=0}=v_0.
\end{equation*}
\notag
$$
Multiplying this equation by $\operatorname{sgn}(v(t))$ and using Kato’s inequality (see [45]), we end up with
$$
\begin{equation}
\frac{d}{dt}\|v(t)\|_{L^1}+\nu|\partial_xv(-\pi)|+ \nu|\partial_xv(\pi)|\leqslant 0.
\end{equation}
\tag{7.37}
$$
Thus, the quantity $\|u_1(t)-u_2(t)\|_{L^1}$ is non-increasing along trajectories. In particular, if we assume that $u_1,u_2\in\mathcal A$, then we infer from (7.37) that and Carleman-type estimates show us that $v(t)\equiv0$ (see [200]). Since, by the standard arguments, equation (7.35) possesses at least one equilibrium, we see that this equilibrium is unique and (7.36) holds. Moreover, the standard compactness arguments show that this equilibrium is exponentially stable, that is,
$$
\begin{equation}
\|u_1(t)-G\|_\Phi\leqslant C_\nu\|u_1(0)-G\|_\Phi\, e^{-\alpha_\nu t}
\end{equation}
\tag{7.38}
$$
for some positive $C_\nu$ and $\alpha_\nu$. Thus, constructing inertial manifolds and studying the limit dynamics on the attractor does not look very interesting for Burgers’ equation (7.35). (Although this equation may still demonstrate a non-trivial intermediate behaviour, which may be interesting from the hydrodynamical point of view (see [17], [141], and the references therein), this phenomenon is more appropriate to study by using the concept of an exponential attractor considered in the next section.) Alternatively, we may return to Burgers’ original model of turbulence, where Burgers’ equation is coupled with an ODE:
$$
\begin{equation}
\frac{d}{dt} U+\nu U=P-\int_{-\pi}^\pi u^2(t,x)\,dx,\qquad \partial_t u-\nu\,\partial_x^2 u+\partial_x(u^2)=Uu
\end{equation}
\tag{7.39}
$$
or to two-component Burgers equations
$$
\begin{equation}
\begin{gathered} \, \frac{d}{dt} U+\nu U=P-\int_{-\pi}^\pi\bigl(u^2(t,x)+v^2(t,x)\bigr)\,dx, \\ \partial_t u-\nu\,\partial_x^2 u+\partial_x(u^2-v^2)=U(u-v),\quad \partial_t v-\nu\,\partial_x^2 v-\partial_x(2uv)=U(u+v), \end{gathered}
\end{equation}
\tag{7.40}
$$
where the function $U(t)$ depends only on $t$ and the components $u$ and $v$ are endowed with Dirichlet boundary conditions (see [22] for the details). In contrast to the single Burgers equation (7.35), we now have instability if the parameter $P>0$ is large enough (this parameter plays the role of external forces), so the attractor, as well as the dynamics on it, becomes non-trivial. On the other hand, we still have an energy identity and dissipativity for these equations, as well as the existence of a smooth absorbing set. For instance, the energy identity for equation (7.40) reads
$$
\begin{equation*}
\frac{1}{2}\,\frac{d}{dt}(U^2+\|u\|^2_{L^2}+\|v\|^2_{L^2})+ \nu(U^2+\|\partial_xu\|^2_{L^2}+\|\partial_xv\|^2_{L^2})=PU.
\end{equation*}
\notag
$$
Moreover, the general theory of inertial manifolds for 1D reaction-diffusion-advection problems is applicable here (we may make the terms containing $\partial_xu$ and $\partial_x v$ small enough to satisfy the spectral gap conditions using the transformation described above). This, in turn, allows us to construct inertial manifolds for both equations (7.39) and (7.40). We observe that this result was stated in [111], however, the proof there was based on an erroneous idea of Kwak, so the correct proof became available only recently due to [132]. We also mention that some particular cases of a general method proposed in [132] were considered in [230]–[232]. Remark 7.16. We recall that the 1D reaction-diffusion-advection equations can be considered as simplified models for the 2D Navier–Stokes equations, so that proving or disproving the existence of inertial manifolds for them was a longstanding open problem of a great theoretical and practical interest. A number of weaker results in this direction have been obtained. We mention here only Romanov’s theory (see [196], [198]), which allowed us to construct Mané projectors with Lipschitz inverses on attractors for some particular cases of these equations (see [127], [197], [243]). A more or less complete answer to this question, which was obtained in [133], [132] and was discussed above, is somehow unexpected and surprising. Indeed, periodic boundary conditions are used in hydrodynamics mainly because they are “simpler” than more physical Dirichlet boundary conditions. The situation with reaction-diffusion-advection equations hints that such an approach may be essentially wrong and, in reality, periodic boundary conditions may be the “most complicated” ones. We conclude our exposition of inertial manifolds by considering the spatial averaging method proposed by Sell and Mallet-Paret and its recent generalizations. We recall that in order to construct an inertial manifold for equation (7.3), we need to solve equation (7.6) backward in time in the properly chosen function space, and the main step here is to solve the corresponding equation of variations (7.16). Moreover, looking at estimate (7.9) we see that the main contribution to the norm (7.8) is made by the intermediate modes which correspond to the eigenvalues satisfying $\lambda_N-k<\lambda_n<\lambda_{N+1}+k$. Let $\mathcal I_{k,N}$ be the spectral orthoprojector onto these modes. Then the most dangerous part of the operator $F'(u(t))$ in (7.16) is exactly its intermediate part $\mathcal I_{k,N}\circ F'(u(t))\circ\mathcal I_{k,N}$ for the properly chosen $k$ and $N$. The key assumption of the method is that this part is close to a scalar operator. Definition 7.17. We say that the nonlinearity $F\in C^1_b(\Phi,\Phi)$ and the operator $A$ satisfy the spatial averaging condition if there exist a bounded measurable function $a\colon\Phi\to\mathbb{R}$ and a positive number $\kappa$ such that, for every $\varepsilon>0$ and every $k\in\mathbb{R}_+$, there exist infinitely many values of $N\in\mathbb N$ satisfying Then the function $a(u)$ is referred to as a spatial averaging of the operator $F'(u)$. Theorem 7.18. Let the operator $A\colon D(A)\to\Phi$, $A=A^*>0$, with compact inverse and nonlinearity $F(u)$ satisfy the spatial averaging condition. Then equation (7.3) possesses an inertial manifold (actually, infinitely many of them) which is the graph of a Lipschitz function over an appropriately chosen spectral subspace. Moreover, if $F$ and its spatial average are more regular, then the corresponding inertial manifold is $C^{1+\varepsilon}$-smooth for some small $\varepsilon>0$. Idea of the proof. We only discuss how to solve the equation of variations (7.16); the full proof can be found in [166] (see also [30], [243]). Let us look at the solution operator $\mathcal L_\theta$ of equation (7.7) for $\theta=(\lambda_{N+1}+\lambda_N)/2$. Then by (7.9) we have
$$
\begin{equation*}
\|\mathcal L_{\theta}\circ (\operatorname{Id}- \mathcal I_{k,N})\|_{\mathcal L(\Phi,\Phi)}\leqslant \frac{1}{k}\,,
\end{equation*}
\notag
$$
and this part of the operator $\mathcal L_\theta$ can be made small by fixing $k$ large enough and taking the standard Lyapunov metric
$$
\begin{equation*}
\|u\|^2_{\Phi,\beta}:=\|(\operatorname{Id}- \mathcal I_{k,N})u\|^2_{\Phi}+\beta\|\mathcal I_{k,N}u\|_{\Phi}^2,
\end{equation*}
\notag
$$
where the parameter $\beta$ is chosen properly, so that we only need to take care about the intermediate modes (see [243] for the details). For these modes we use the spatial averaging condition, which allows us to reduce the problem to equations of the form Finally, the scalar term $a(u(t))v_n$ can be removed by the change of independent variables $v_n(t)=\exp\biggl\{\displaystyle\int_0^sa(u(s))\,dx\biggr\}w_n(t)$. It is important that this transformation is the same for all (intermediate) modes. This reduces the problem to and this problem can be solved using the fact that $\lambda_{N+1}-\lambda_N>\kappa$.
This finishes the proof of solvability for the equation of variations. The transition to the nonlinear case is a bit more complicated here since the weight $\exp\biggl\{\displaystyle\int_0^ta(u(s))\,ds\biggr\}$ depends on the trajectory $u(t)$, which is also unknown from the very beginning. For this reason methods related to invariant cones are traditionally used here (see [166], [243], [30] for the details), although this transition can also be done by means of perturbation arguments and Banach’s contraction theorem (see [126]). In applications the operator $A$ is usually the minus Laplacian with periodic boundary conditions, mainly in the 3D case, say, in $\Phi=L^2(\Omega)$, where $\Omega=[-\pi,\pi]^3$ (see [166], [146] for applications to irrational tori in 2D or/and equilateral triangles). In this case the eigenvalues $\lambda_n$ are naturally parameterized by points in the lattice $\mathbb Z^3$:
$$
\begin{equation*}
\lambda_{\vec n}=m^2+l^2+k^2,\qquad \vec n=(m,l,k)\in\mathbb Z^3,
\end{equation*}
\notag
$$
and the number theory related to distributions of sums of squares comes into play in a natural way. Namely, assume in addition that the operator $F'(u)$ is pointwise multiplication, by a sufficiently smooth function $\psi=\psi_u(x)$ (or is a sum of combinations of pointwise multiplication and differentiation). Then, with respect to the Fourier basis, the action of $F'(u)$ becomes a convolution:
$$
\begin{equation}
(F'(u)v)_{\vec n}:=\sum_{\vec n_1\in \mathbb Z^3}\psi_{\vec n-\vec n_1}v_{\vec n_1}
\end{equation}
\tag{7.42}
$$
and we may use the following lemma to verify the spatial averaging condition. Lemma 7.19. Let Then, for every $k>0$ and $r>0$, there exist infinitely many $N\in\mathbb N$ such that The proof of this lemma is given in [166]. This lemma shows that the intermediate part of the operator $F'(u)v$ contains the leading term $\psi_0 v$, and all other terms contain $\psi_{\vec n}$ with $|\vec n|>\rho$ (which are small if $\psi$ is smooth enough). Thus, the spatial averaging condition is satisfied for $a(u)=\psi_0=\langle \psi_u\rangle $, and this justifies the interpretation of $a(u)$ as a spatial averaging in the general theory (see [166], [243], and the references therein for more details). This scheme was proposed by Sell and Mallet-Paret [166] to construct an inertial manifold for a scalar reaction-diffusion equation (7.12) in 3D with periodic boundary conditions. Clearly, the spectral gap conditions are not satisfied there, and this was historically the first example of an inertial manifold beyond the spectral gap conditions. They also gave a counterexample showing that in 4D this method will not work and the inertial manifold may not exist (at least in the class of normally hyperbolic inertial manifolds; see [167]). The corresponding counterexample for a system of two reaction-diffusion equations in 3D (also with periodic boundary conditions) where an inertial manifold does not exist (again, in the class of normally-hyperbolic inertial manifolds) was given by Romanov [197]. The inertial manifolds for the Cahn–Hilliard equations in 3D periodic domains were constructed in [131] by means of a similar method. As we have already mentioned, the spatial averaging method does not work in general for systems of PDEs. Indeed, in this case $a(u)$ will not be a scalar operator, but a matrix operator, and this is not enough to solve the simplified equation of variations for the intermediate modes. One exception is the case where $a(u)$ is a zero matrix: then everything should work similarly to the scalar case. However, there was a non-trivial problem here related to a cut-off of the initial equation transforming it into the so-called “prepared” form. Usually, the spatial averaging method requires a very non-trivial cut-off procedure, and it was not clear how to preserve the condition $\langle a(u)\rangle=0$ under this procedure. The progress here is mainly related to Kostianko’s paper [126], where such a cut-off procedure was proposed, which uses an appropriate modification of the nonlinearity $F(u)$ in the Fourier basis. In turn, this construction allowed us to use spatial averaging method for inertial manifolds for some hydrodynamical problems. For instance, the existence of an inertial manifold for the modified Leray-$\alpha$ model and hyperviscous Navier–Stokes equations on a 3D torus was proved in [126] and [82], respectively, and more general problems of the form
$$
\begin{equation}
\begin{cases} \partial_t u+(u,\nabla_x\overline u)+(-\Delta_x)^{1+\gamma}u+\nabla_x p=g,& u\big|_{t=0}=u_0, \\ \operatorname{div} u=0,& \overline u=(1-\alpha\Delta_x)^{-\overline\gamma}u, \end{cases}
\end{equation}
\tag{7.45}
$$
where $\gamma,\overline\gamma\geqslant 0$ and $\gamma+\overline\gamma=1/2$, were considered in [30]. Let us now return to the case of systems where $a(u)$ is not a scalar matrix and consider the following version of a cross-diffusion system:
$$
\begin{equation}
\partial_t u-(1+i\omega)(\Delta_x u-u)=f(u,\overline u),\quad u\big|_{t=0}=u_0,
\end{equation}
\tag{7.46}
$$
where $u(t,x):=\operatorname{Re} u(t,x)+i\operatorname{Im} u(t,x)$ is a complex-valued unknown function, $\overline u$ is the complex conjugate, $f$ is a given smooth function with finite support, and $\omega\ne0$ is a real cross-diffusion parameter. We consider this equation on the 3D torus $\Omega=[-\pi,\pi]^3$. The typical example of such an equation is the complex Ginzburg–Landau equation
$$
\begin{equation}
\partial_t u-(1+i\omega)(\Delta_x u-u)=\varphi(|u|^2)u.
\end{equation}
\tag{7.47}
$$
As usual, we assume that this equation possesses a smooth absorbing set, so without loss of generality we may assume that $\varphi$ is smooth and has a finite support. Applying the method of spatial averaging to the equation of variations related to problem (7.46), we end up with the following problem for the intermediate modes:
$$
\begin{equation}
\frac{d}{dt}v_n+(1+i\omega)(\lambda_n+1)v_n=\langle f'_u\rangle v_n+ \langle f'_{\overline u}\rangle \overline v_n+\text{``small''}.
\end{equation}
\tag{7.48}
$$
We see that the first term on the right-hand side is a nice scalar operator, but the second ($\langle f'_{\overline u}\rangle \overline v_n$) is not, so the sole spatial averaging is not sufficient for constructing an inertial manifold (the corresponding example with $\omega=0$ was given in [197]). The key idea here is (following [127]) to combine the spatial averaging method with temporal averaging with respect to the rapid in time oscillations generated by the large dispersion term $i\omega(\lambda_n+1)v_n$ in equations (7.48). This averaging kills (makes small) the non-scalar term $\langle f'_{\overline u}\rangle \overline v_n$ in (7.48) and allows us to complete the proof of the existence of an inertial manifold for equation (7.46) in the case where $\omega\ne0$ (see [127] and [130] for the missing details).
8. Exponential attractors and perturbation theoryWe have seen that inertial manifolds gives us a perfect way to constructing a finite-dimensional reduction for the limit dissipative dynamics. However, an inertial manifold requires much more restrictive conditions to exist than an attractor. In this section we discuss an intermediate concept between attractors and inertial manifolds which was introduced in [63], namely, the concept of an exponential attractor, which, on the one hand, is almost as common as usual attractors and, on the other hand, allows us to overcome the most principal drawbacks of usual (global) attractors. We start with an illustrative example. Example 8.1. Consider the following 1D real Ginzburg–Landau equation with non-zero boundary conditions: Thus, the limit dynamics is trivial for any $\nu>0$. It is just a single exponentially stable equilibrium. Moreover, linearizing our equation near $u=1$ we see that the rate of attraction to the equilibrium is not slower than $e^{-2t}$ (and does not become worse as $\nu\to0$). So we may conclude that, for any (bounded) set $B\subset\Phi$,
$$
\begin{equation}
\operatorname{dist}_\Phi(S(t)B,\mathcal A)\leqslant Ce^{-2t},
\end{equation}
\tag{8.3}
$$
where the constant $C$ is actually independent of $B$, but may still depend on $\nu$. However, for small $\nu>0$, equation (8.1) possesses an interesting metastable dynamics related to the slow evolution of multi-kinks. For instance, if you start from a profile that is close to where $H(x)$ is the standard Heaviside function, then the corresponding solution is close to $u_0(x)$ for an extremely long period of time: the life-span of this “almost” equilibrium $T$ is asymptotically $ e^{-1/\nu}$ (see [61], [245] for more details about the evolution of multi-kinks via the centre manifold reduction). Thus, for the constant $C$ in (8.3), we have an estimate with some positive $c$ which is independent of $\nu$.Thus, the behaviour of trajectories of (8.1) for small $\nu$ is described mainly by the evolution of metastable states, and the attractor $\mathcal A$ becomes unobservable in experiment and has only a limited theoretical interest. Remark 8.2. The situation described above, where the limiting behaviour is trivial, but there is a rich and interesting metastable dynamics, is somehow typical for dissipative systems and is observed in many other situations, for instance in the 1D Burgers equation (7.35) (see [17], [141], and the references therein). A similar situation also arises for the stochastic equation (5.21) for small values of $\varepsilon>0$, where the corresponding random attractor is trivial (we even have the uniform estimate (5.26) for the Lyapunov exponent), but it takes a very long time (depending on $\varepsilon$ in a way similar to (8.4)) to reach this attractor. To avoid such problems, it is crucial to control the rate of attraction to the attractor under consideration in terms of physical parameters. Unfortunately, there are no ways to get this control on the level of usual (global) attractors in a more or less general situation. On the other hand, we often have such a control for the rate of attraction to an inertial manifold, which allows us to capture the metastable dynamics as well. Capturing this dynamics in more general cases, where an inertial manifold may not exist, is one of the main sources of motivation for the theory of exponential attractors. Actually, in the examples discussed above, we have an exponential rate of attraction to the corresponding global attractors, so the most important advantage of the theory of exponential attractors (at least in these cases) is exactly the ability to get a reasonable control (not like (8.4)) for the exponential rate of attraction in terms of physical parameters. The key idea of an exponential attractor is to add the metastable states to the global attractor so that, on the one hand, to get an effective control of the rate of attraction and, on the other hand, to preserve finite-dimensionality (in terms of the fractal dimension), which allows us to use Mané’s projection theorem. Definition 8.3. Let $S(t)\colon\Phi\to\Phi$ be a dynamical system acting on a metric space $\Phi$ with a fixed bornology $\mathbb B$. A set $\mathcal M$ is an exponential attractor for the dynamical system $S(t)$ if $\mathcal M$ satisfies the following conditions: 1) it is compact in $\Phi$ and has a finite fractal dimension: 2) it is semi-invariant, that is, $S(t)\mathcal M\subset\mathcal M$ for $t\geqslant 0$; 3) it attracts exponentially the images of $B\in\mathbb B$: for some positive constants $\alpha$ and $Q_B$.We recall some basic facts of perturbation theory for global attractors before discussing the constructions of exponential attractors. We start with upper semicontinuity, which is based on the following fact from general topology. Proposition 8.4. Let $\Phi$ and $\mathfrak A$ be two Hausdorff topological spaces, and let $\mathbb A$ be a compact set in $\Phi\times\mathfrak A$. Also let $\Pi_1$, $\Pi_2$ be the projectors of $\mathbb A$ onto the first and second component of the Cartesian product, and let $\mathcal A_\alpha:=\Pi_1\Pi_2^{-1}(\alpha)$, $\alpha\in\mathfrak A$. Then the family of sets $\{\mathcal A_\alpha\}_{\alpha\in\mathfrak A}$ is upper semicontinuous at every $\alpha_0\in\mathfrak A$, namely, for every neighbourhood $\mathcal O(\mathcal A_{\alpha_0})$ of the set $\mathcal A_{\alpha_0}$ in $\Phi$ there exists a neighbourhood $\mathcal O(\alpha_0)$ in $\mathfrak A$ such that The proof of this proposition can be found, for example, in [12] for the case where $\Phi$ and $\mathfrak A$ are metric spaces. The general case can be dealt with analogously. Note also that in the metric case upper semicontinuity can be rewritten in an equivalent way using the Hausdorff semi-distance:
$$
\begin{equation}
\lim_{\alpha\to\alpha_0}\operatorname{dist}_{\Phi} (\mathcal A_\alpha,\mathcal A_{\alpha_0})=0.
\end{equation}
\tag{8.8}
$$
In applications, we usually take $\mathbb A:=\bigcup\limits_{\alpha\in \mathfrak A}\mathcal A_\alpha\times\{\alpha\}$, where the $\mathcal A_\alpha$ are the attractors of dynamical systems $S_\alpha(t)\colon\Phi\to\Phi$ depending on the parameter $\alpha\in\mathfrak A$. Then, to verify upper semicontinuity, we just need to consider an arbitrary sequence $u_n^0\in \mathcal A_{\alpha_n}$, where $\alpha_n\to\alpha_0$, and extract from it a subsequence $u_{n_k}^0$ converging to some $u^0_{\alpha_0}\in\mathcal A_{\alpha_0}$. In turn, in order to check this property, we may use the representation formula for $\mathcal A_{\alpha_n}$, so we only need to consider a sequence of complete bounded solutions $u_{\alpha_n}(t)\in\mathcal K_{\alpha_n}$ and extract a subsequence $u_{\alpha_n}(t)$ converging to $u_{\alpha_0}(t)\in\mathcal K_{\alpha_0}$. This is usually true under the minimal assumptions on the dynamical system $S_\alpha(t)$. We emphasize that this method does not require one to verify the closeness of individual semi-trajectories of the perturbed and non-perturbed dynamical systems (which is often a much more difficult task, especially in the case of singular perturbations) and is applicable, for example, to trajectory attractors (see [12], [38], [179], and the references therein). In contrast to upper semicontinuity, the lower semicontinuity of attractors $\mathcal A_\alpha$, $\alpha\in\mathfrak A$, can easily be broken (see Example 2.5). Recall that in the metrizable case, the lower semicontinuity of the family $\{\mathcal A_{\alpha}\}_{\alpha\in\mathfrak A}$ reads
$$
\begin{equation}
\lim_{\alpha\to\alpha_0}\operatorname{dist}_\Phi (\mathcal A_{\alpha_0},\mathcal A_\alpha)=0,\qquad \alpha_0\in\mathfrak A.
\end{equation}
\tag{8.9}
$$
The continuity of the family of attractors $\{\mathcal A_\alpha\}_{\alpha\in\mathfrak A}$ means that (8.8) and (8.9) hold simultaneously, that is,
$$
\begin{equation}
\lim_{\alpha\to\alpha_0}\operatorname{dist}_\Phi^{\rm sym} (\mathcal A_\alpha,\mathcal A_{\alpha_0})=0,\qquad \alpha_0\in\mathfrak A,
\end{equation}
\tag{8.10}
$$
where $\operatorname{dist}_\Phi^{\rm sym}(U,V):= \max\{\operatorname{dist}_\Phi(U,V),\operatorname{dist}_\Phi(V,U)\}$ is the symmetric Hausdorff distance between sets. The next standard proposition shows that the continuity of attractors is determined by the rate of attraction. Proposition 8.5. Let $\Phi$ and $\mathfrak A$ be subsets of normed spaces, and let $S_\alpha(t)\colon\Phi\to\Phi$ be a family of dynamical systems on $\Phi$ satisfying for some positive constants $C$ and $K$ which are independent of $\alpha_i\in\mathfrak A$ and $u_0\in\Phi$. Assume also that $\mathfrak A$ is compact, the $S_\alpha(t)$ are continuous for every fixed $t$ and $\alpha$, and, for every $\alpha\in\mathfrak A$, the dynamical system $S_\alpha(t)$ possesses an attractor $\mathcal A_\alpha$ with respect to the bornology of all subsets of $\Phi$. Then 1) the family $\{\mathcal A_\alpha\}_{\alpha\in\mathfrak A}$ is uniformly continuous with respect to $\alpha\in\mathfrak A$ (that is, (8.10) holds uniformly with respect to $\alpha_0$) if and only if
$$
\begin{equation}
\lim_{t\to\infty}\,\sup_{\alpha\in\mathfrak A}\,\operatorname{dist}_\Phi (S_\alpha(t)\Phi,\mathcal A_\alpha)=0;
\end{equation}
\tag{8.12}
$$
2) if, in addition, the following stronger version of (8.12) (exponential attraction) holds: where the positive constants $Q$ and $\lambda$ are independent of $t$ and $\alpha$, then the family of attractors $\mathcal A_\alpha$ is uniformly Hölder continuous: Proof. We consider here only the second part of the statement; the first part is more straightforward (see, for example, [12], [93], [101], [154]). Indeed, let $u_{\alpha_1}\in\mathcal A_{\alpha_1}$. Then, by the invariance of the attractor, for any $T>0$ there exists $v_{\alpha_1}\in\mathcal A_{\alpha_1}$ such that $S_{\alpha_1}(T)v_{\alpha_1}=u_{\alpha_1}$. Consider the point $\overline u_{\alpha_1}:=S_{\alpha_2}(T)v_{\alpha_1}$. Then, due to (8.13), we have
$$
\begin{equation*}
\operatorname{dist}_\Phi(\overline u_{\alpha_1},\mathcal A_{\alpha_2}) \leqslant Q\,e^{-\lambda T}
\end{equation*}
\notag
$$
and, due to (8.11), we know that
$$
\begin{equation*}
\|u_{\alpha_1}-\overline u_{\alpha_1}\|_\Phi\leqslant C\|\alpha_1-\alpha_2\|_{\mathfrak A}\,e^{KT}.
\end{equation*}
\notag
$$
Therefore, using that $u_{\alpha_1}\in\mathcal A_{\alpha_1}$ is arbitrary, we have
$$
\begin{equation*}
\operatorname{dist}_\Phi(\mathcal A_{\alpha_1},\mathcal A_{\alpha_2}) \leqslant Qe^{-\lambda T}+C\|\alpha_1-\alpha_2\|_{\mathfrak A}\,e^{KT}.
\end{equation*}
\notag
$$
Finally, fixing $T=\dfrac{1}{\lambda+K}\ln\dfrac{1}{\|\alpha_1-\alpha_2\|_{\mathfrak A}}$, we arrive at
$$
\begin{equation*}
\operatorname{dist}_\Phi(\mathcal A_{\alpha_1},\mathcal A_{\alpha_2})\leqslant (Q+C) \|\alpha_1-\alpha_2\|_{\mathfrak A}^\kappa,\qquad \kappa:=\frac{\lambda}{K+\lambda}\,.
\end{equation*}
\notag
$$
Swapping $\alpha_1$ and $\alpha_2$, we get the desired estimate (8.14). Remark 8.6. Assumption (8.11) may look too restrictive since it requires uniformity with respect to all $u_0\in\Phi$, but it is actually only necessary for $u_0\in\mathcal A:=\bigcup\limits_{\alpha\in\mathfrak A}\mathcal A_{\alpha}$. The uniform attraction property is also necessary for the set $\mathcal A$ only, so the assumption that the bornology on $\Phi$ consists of all subsets of $\Phi$ is not restrictive at all. We may also formulate a natural non-uniform analogue of Proposition 8.5 for the upper and lower semicontinuity of attractors $\mathcal A_\alpha$ at a single point $\alpha=\alpha_0$, as well as its analogues for the non-autonomous case (see [93] for more details). Crucial for us is that the continuity of attractors $\mathcal A_\alpha$ is determined by the rate of attraction, and even qualitative bounds for the closeness of the perturbed and non-perturbed attractors are automatically obtained if the rate of attraction is under control. Unfortunately, the problem of getting this control looks unsolvable on the level of usual (global) attractors in a more or less general situation. This makes attractors somehow unobservable, namely, no matter how long you observe the system and how precise your measurements/simulations are, you cannot guarantee that the reconstructed approximate attractor is close to the precise one, and this is an extra source of motivation to consider exponential attractors. Note also that the Hölder continuity (8.14) is the best what we can expect and it cannot be improved to Lipschitz continuity even when ideal objects like Morse–Smale or uniformly hyperbolic attractors are considered (see [116] and the references therein). Remark 8.7. We also mention an interesting result, which shows that the lower continuity of attractors $\mathcal A_\alpha$ holds for “typical” values of the parameter $\alpha\in\mathfrak A$ (see [101] and [233]). This result tells us that, given a family of dynamical systems $S_\alpha(t)$ (acting on a separable, complete, and bounded metric space $\Phi$) which depends continuously on both $t$ and $\alpha\in\mathfrak A$, where $\mathfrak A$ is a compact metric space, the corresponding attractors $\mathcal A_\alpha$ (if exist) are upper and lower semicontinuous for every $\alpha$ belonging to a residual subset of $\mathfrak A$ (that is, for every $\alpha\in\mathfrak A$ away from a countable union of nowhere dense sets). Moreover, given a Borel probability measure $\mu$ on $\mathfrak A$, for every $\varepsilon>0$ there exists a closed set $\mathfrak A_\varepsilon$ satisfying $\mu(\mathfrak A\setminus\mathfrak A_\varepsilon)\leqslant\varepsilon$, such that the family $\mathcal A_\alpha$ of attractors is uniformly continuous on $\mathfrak A_\varepsilon$ in the sense of the symmetric Hausdorff distance. Due to the first part of Proposition 8.5, this, in turn, gives us a uniform rate of attraction to the attractors $\mathcal A_\alpha$ with respect to $\alpha\in\mathfrak A_\varepsilon$. Indeed, consider the space $B\Phi$ of all closed non-empty subsets of $\Phi$, with the symmetric Hausdorff distance as a metric. Then it known that $B\Phi$ is a complete separable metric space and the continuity of the family $\mathcal A_\alpha$ can be interpreted as the continuity of the map $f\colon\mathfrak A\to B\Phi$ defined by $f(\alpha):=\mathcal A_\alpha$. The idea of the proof consists in approximating the function $f(\alpha)$ by $f_n(\alpha):=S_\alpha(n)\Phi$. It is not difficult to see that the continuity assumptions imposed on $S_\alpha(t)$ imply that the functions $f_n$ are continuous and the existence of the attractors $\mathcal A_\alpha$ ensures that $f_n(\alpha)\to f(\alpha)$ pointwise. Thus, by Baire’s theorem, the set of discontinuities of $f(\alpha)$ is a countable union of nowhere dense sets. The second statement is an immediate corollary of Egorov’s theorem. It is worth mentioning that these results cannot replace exponential attractors since they neither give any reasonable way to find the set $\mathfrak A_\varepsilon$ explicitly or compute it, nor any way to control the uniform rate of attraction to $\mathcal A_\alpha$ for $\alpha\in\mathfrak A_\varepsilon$ in terms of the physical parameters. The result of Proposition 8.5 can partially be extended to exponential attractors. Proposition 8.8. Let the assumptions of Proposition 8.5 hold, and let the dynamical system $S_\alpha(t)$ possess exponential attractors $\mathcal M_\alpha$, $\alpha\in\mathfrak A$ which satisfy the attraction property (8.13) uniformly with respect to $\alpha\in\mathfrak A$. Then the following estimate holds: The proof of this fact repeats word by word the arguments given in the proof of Proposition 8.5, and for this reason it is omitted. Remark 8.9. The robustness of exponential attractors with respect to perturbations was stated in the original work [63] exactly in the form (8.15). The reason why we have to replace $\mathcal M_\alpha$ by $\mathcal A_\alpha$ in (8.15) is that an exponential attractor is only semi-invariant, so starting from $u_\alpha\in \mathcal M_\alpha$, we may fail to find $v_\alpha$ in $\mathcal M_\alpha$ (or in a set satisfying the uniform attraction property) such that $S_\alpha(T)v_\alpha=u_\alpha$. In fact, we can do this if $u_\alpha\in S_\alpha(t)\mathcal M_\alpha$ for $t$ large enough, so the following shifted version of Hölder continuity holds: We state below the transitivity of exponential attraction, which is one of the key tools in the theory of exponential attractors. Proposition 8.10. Let $\Phi$ be a metric space and $S(t)$ be a dynamical system on it which is Lipschitz continuous: The proof of estimate (8.18) is based on arguments which are similar to the derivation of (8.14) and can be found in [72]. 8.1. Exponential attractors via squeezing propertiesWe now turn to theorems which guarantee the existence of exponential attractors with nice properties. These theorems are based on various forms of the squeezing property and are very close to the ones presented in subsection 6.2. Namely, the iterative $\varepsilon$-nets constructed in order to estimate the fractal dimension of an attractor are exactly the “metastable” states which we need to add to the attractor in order to get exponential attraction. We demonstrate the main idea using an analogue of Theorem 6.4 that was proved in [65] (see also [71] and [179] for more details). Theorem 8.11. Let $\Phi$ and $\Phi_1$ be two Banach spaces and let the embedding $\Phi_1\subset\Phi$ be compact. Assume also that we are given a bounded subset $\mathcal B$ of $\Phi_1$ and a map $S\colon\mathcal B\to\mathcal B$ satisfying the squeezing property Then the discrete dynamical system $S(n):=S^n$, $n\in\mathbb N$, possesses an exponential attractor $\mathcal M$ with the following properties: 1) $\mathcal M$ is compact in $\Phi_1$ and is semi-invariant: $S\mathcal M\subset\mathcal M$; 2) its fractal dimension in $\Phi_1$ is finite and satisfies the estimate
$$
\begin{equation}
\dim_{\rm f}(\mathcal M,\Phi_1)\leqslant \mathbb H_{1/(4L)}(\Phi_1\hookrightarrow\Phi);
\end{equation}
\tag{8.20}
$$
3) $\mathcal M$ attracts the set $\mathcal B$ exponentially and where $R_0$ is such that $\mathcal B\subset B_{R_0}(0,\Phi)$. Sketch of the proof. Indeed, arguing as in the proof of Theorem 6.4, for every $n\in\mathbb N$ we construct an $R_0\,2^{-n}$-net in $V_n\subset S(n)\mathcal B$ such that $V_n\subset S(n)\mathcal B$ and $\#V_n\leqslant N^{n}$ where $N$ is the same as in the proof of Theorem 6.4. To preserve semi-invariance we set $E_1=V_1$ and $E_n=V_n\cup S(E_{n-1})$ for $n=2,3,\dots$ . Then $E_n\subset S(n)\mathcal B$, $\#E_n\leqslant CN^{n+1}$, and $S(E_n)\subset E_{n+1}$. Let us finally define
$$
\begin{equation}
\mathcal M'=\bigcup_{n\in\mathbb N}E_n,\quad \mathcal M:=[\mathcal M']_{\Phi_1}.
\end{equation}
\tag{8.22}
$$
Then semi-invariance, as well as the exponential attraction (8.21), hold immediately and we only need to check estimate (8.20) for the fractal dimension. Let $\varepsilon_n=R_0\,2^{-n}$. Then all the sets $E_k$ for $k\geqslant n$ are subsets of $S(n)\mathcal B$, and therefore the $\varepsilon_n$-balls centred at points of $V_n$ cover them. So the set $\bigcup\limits_{k=1}^nE_k$ is an $\varepsilon_n$-net of $\mathcal M$, and therefore
$$
\begin{equation*}
\begin{aligned} \, \limsup_{n\to\infty}\frac{\mathbb H_{\varepsilon_n}(\mathcal M,\Phi_1)} {\log_2(1/\varepsilon_n)}&\leqslant \lim_{n\to\infty} \frac{\log_2 C+\log_2(\sum_{k=1}^n\#E_n)}{n-\log_2R_0} \\ &\leqslant \lim_{n\to\infty}\frac{2\log_2C+(n+2) \log_2N}{n-\log_2R_0}=\log_2N, \end{aligned}
\end{equation*}
\notag
$$
which gives the desired estimate (8.20) and finishes the proof of the theorem. Now assume that we are given a continuous dynamical system $S(t)$ in $\Phi$ such that $S=S(1)$ satisfies all the assumptions of Theorem 8.11. Then, first, we can construct a discrete exponential attractor $\mathcal M_d$ (that is, an attractor for the semigroup $S(n)$, $n\in\mathbb N$) and after that can extend it to a continuous one via
$$
\begin{equation}
\mathcal M:=\biggl[\,\bigcup_{t\in[1,2]}S(t)\mathcal M_d\biggr]_{\Phi}.
\end{equation}
\tag{8.23}
$$
Indeed, all the properties of an exponential attractor, except of the control of fractal dimension, follows automatically from the analogous properties of $\mathcal M_d$, but to control the fractal dimension in $\Phi$, we need an extra Hölder continuity assumption:
$$
\begin{equation}
\|S(t_1)u_1-S(t_2)u_2\|_\Phi\leqslant C\bigl(|t_1-t_2|+\|u_1-u_2\|_\Phi\bigr)^\alpha,\qquad t_1,t_2\in[1,2],\quad u_1,u_2\in\mathcal B,
\end{equation}
\tag{8.24}
$$
for some positive $C$ and $\alpha$. Then a transition from a discrete to a continuous exponential attractor can increase the fractal dimension in $\Phi$ by the additive factor $\alpha^{-1}$ at most. Moreover, using that $S(1)\mathcal M$ is also an exponential attractor if $\mathcal M$ is, we may also get the finiteness of the fractal dimension in $\Phi_1$ (due to (8.19), the fractal dimension of $S(t)\mathcal M$ in $\Phi_1$ is controlled by the fractal dimension of‘$\mathcal M$ in $\Phi$). Remark 8.12. In applications $\mathcal B$ is usually a compact absorbing or an exponentially attracting set of the semigroup $S(t)$. Then, due to the transitivity of exponential attraction, we conclude that the attractor $\mathcal M$ constructed attracts not only the set $\mathcal B$, but all bounded sets in the phase space $\Phi$. Note also that for semigroups $S(t)$ generated by PDEs, the uniform Hölder continuity in time stated in (8.24) typically holds only if $\mathcal B$ is more smooth than the initial phase space $\Phi$. This is not a problem when parabolic PDEs are considered since, due to the instantaneous smoothing property, we may find a compact (and more regular) absorbing set $\mathcal B$. However, in more general cases (for example, for damped wave equations), we only have an asymptotic smoothing property, so $\mathcal B$ must be a compact exponentially attracting set and the use of the transitivity of exponential attraction becomes unavoidable (see [72], [179], and the references therein). At the next step, following [71] we consider an analogue of Theorem 6.8, which also includes the Hölder continuity of exponential attractors with respect to perturbations. To this end, we need the following definition. Definition 8.13. Let $\Phi_1\subset\Phi$ be two Banach spaces such that the embedding is compact, and let a bounded subset $\mathcal B$ of $\Phi_1$, $\varepsilon>0$, $\kappa\in[0,1)$, and $L>0$ be given. Then a map $S\colon\mathcal O_\varepsilon(\mathcal B)\to\mathcal B$ belongs to the class $\mathbb S_{\varepsilon,\kappa,L}(\mathcal B)$ if the squeezing property (6.10) is satisfied for all $u_1,u_2\in \mathcal O_\varepsilon(\mathcal B)$. The distance between two maps $S_1,S_2\in\mathbb S_{\varepsilon,\kappa,\delta}(\mathcal B)$ is defined as follows: Theorem 8.14. Let $S\in\mathbb S_{\varepsilon,\delta,L}(\mathcal B)$, and let $S(n)=S^n$ be a discrete dynamical system on $\mathcal B$ associated with this map. Then this semigroup possesses an exponential attractor $\mathcal M_S\subset \mathcal B$, which has the following properties: 1) $\mathcal M_S$ is a compact semi-invariant set in $\mathcal B$ whose fractal dimension satisfies an analogue of estimate (6.11); 2) the following exponential attraction property holds:
$$
\begin{equation}
\operatorname{dist}_{\Phi_1}(S(n)\mathcal B,\mathcal M_S)\leqslant \varepsilon\biggl(\frac{1+\kappa}2\biggr)^n;
\end{equation}
\tag{8.25}
$$
3) if $S_1,S_2\in\mathbb S_{\varepsilon,\kappa,L}(\mathcal B)$, then for some positive $C$ and $\theta$ depending only on $\varepsilon$, $\kappa$, $L$, and the spaces $\Phi$ and $\Phi_1$. Idea of the proof. The attractors $\mathcal M_S$ can be constructed using the $\varepsilon\biggl(\dfrac{\kappa+1}2\biggr)^n$-nets $E_n=E_n(S)$ similarly to Theorem 8.11 (see also Theorem 6.8), so we only need to explain how to obtain Hölder continuity (8.26). We actually need to estimate the distance between $E_n(S_1)$ and $\mathcal M_{S_2}$ for all $n\in\mathbb N$. For large values of $n$ we use (8.16) and get the desired estimate without extra care. In contrast to this, for relatively small values of $n$ we need the extra assumption that the sets $E_n$ are constructed in such a way that
$$
\begin{equation}
\operatorname{dist}_{\Phi_1}^\mathrm{sym}(E_n(S_1),E_n(S_2))\leqslant K^n \|S_1-S_2\|_{\mathbb S}
\end{equation}
\tag{8.27}
$$
for some $K$ which is independent of $n$, $S_1$, and $S_2$. The details can be found in [71]. Remark 8.15. Recall that, similarly to inertial manifolds, exponential attractors are not unique, so the problem of choosing an “optimal” exponential attractor becomes crucial for both theory and applications. The theorem stated above gives us a single-valued Hölder continuous branch of the function $S\to\mathcal M_S$ for wide class of nonlinear maps $S\in\mathbb S_{\varepsilon,\kappa,L}(\mathcal B)$. We now turn to non-autonomous dynamical systems and start with the uniform (deterministic) case. We say that a discrete cocycle $S_\xi(n)\colon\mathcal O_\varepsilon(\mathcal B)\to\mathcal B$, $n\in\mathbb N$, over a dynamical system $T(n)\colon\Psi\to\Psi$ belongs to the class $\mathbb S_{\varepsilon,\kappa,L}(\mathcal B)$ if $S_\xi(1)\in\mathbb S_{\varepsilon,\kappa,L}(\mathcal B)$ for all $\xi\in\Psi$. Then the non-autonomous analogue of Theorem 8.14 reads as follows. Theorem 8.16. Let the cocycle $S_\xi(n)$ belong to the class $\mathbb S_{\varepsilon,\kappa,L}(\mathcal B)$, and let $U_\xi(m,n)$, $m\geqslant n$, $\xi\in\Psi$, be the corresponding dynamical processes on $\mathcal B$. Then there exists a family $\mathcal M_S(\xi)$, $\xi\in\Psi$, of compact sets in $\mathcal B$ (a non-autonomous exponential attractor) which has the following properties. 1) Their fractal dimensions are finite and uniformly bounded:
$$
\begin{equation*}
\dim_{\rm f}(\mathcal M_S(\xi),\Phi_1)\leqslant C,\qquad \xi\in\Psi.
\end{equation*}
\notag
$$
2) They are semi-invariant: $S_\xi(1)\mathcal M_S(\xi)\subset \mathcal M_S(T(1)\xi)$, $\xi\in\Psi$, and the following uniform exponential attraction property holds:
$$
\begin{equation}
\operatorname{dist}_{\Phi_1}(S_\xi(n)\mathcal B,\mathcal M_S(T(n)\xi)) \leqslant Qe^{-\alpha n},\qquad \xi\in\Psi,\quad n\in\mathbb N,
\end{equation}
\tag{8.28}
$$
for some positive constants $Q$ and $\alpha$ depending only on $\Phi$, $\Phi_1$, $\mathcal B$, $\varepsilon$, $\kappa$, and $L$. 3) For any two cocycles $S_\xi(n)$ and $\hat S_\xi(n)$ over the same dynamical system $T(n)\colon\Psi\to\Psi$ belonging to the class $\mathbb S_{\varepsilon,\kappa,L}(\mathcal B)$, the following uniform Hölder continuity holds: where the positive constants $C$, $\beta$, and $\theta$ depend only on $\Phi$, $\Phi_1$, $\mathcal B$, $\varepsilon$, $\kappa$, and $L$.The proof of this result is almost identical to the proof of Theorem 8.14 and is given in [71]. Remark 8.17. As usual, the passage from discrete to continuous time requires some uniform Hölder regularity with respect to time for the cocycle under consideration (an analogue of estimate (8.24)). In this case the desired exponential attractor $\mathcal M_S(\xi)$ can be defined via As mentioned already, the assumption of Hölder continuity in time may be rather restrictive, since it usually requires extra regularity in space (although in most applications it is not a big problem due to the transitivity of exponential attraction), so it would be interesting to relax it. The attempts to do so are related to the use of a more straightforward and naive (than (8.30)) extension, namely, the continuous attractor is defined via $\mathcal M_S(t):=\mathcal M_S(T(t)\xi)$ for $t=n\in\mathbb Z$ and, for $t=n+s$, where $n\in\mathbb Z$ and $s\in[0,1)$, we set $\mathcal M_S(t):=S_{T(n)\xi}(s)\mathcal M_S^d(n)$. In this case, we indeed have semi-invariance, exponential attraction, and uniform bounds for the fractal dimension without Hölder continuity in time, but the object obtained will no longer be compatible with the autonomous case (where it will give an artificial time-periodic attractor), it will not be continuous in time (artificial jumps at integer points), and so on. Since such an object can hardly be considered as a satisfactory version of a non-autonomous exponential attractor, the problem of removing/relaxing Hölder continuity in time remains open. Remark 8.18. We emphasize that the non-autonomous exponential attractor $\mathcal M_S(\xi)$, $\xi\in\Psi$, constructed in Theorem 8.16 is not just a pullback attractor: it also attracts forward in time at a uniform exponential rate. This demonstrates one of the main advantages of exponential attractors for non-autonomous equations, namely, they allow us to settle the problem with forward attraction, which looks unsolvable on the level of pullback attractors (see Example 5.10) and, in contrast to uniform attractors, the object constructed remains finite-dimensional. For this reason, the name of “pullback” exponential attractors, which is used by some authors (see, for example, [28]), looks confusing for us, and we prefer to refer to them as non-autonomous exponential attractors. Also, in contrast to uniform attractors, such exponential attractors do not violate the causality principle. Indeed, estimate (8.29) shows us that the exponential attractor $\mathcal M_S(\xi)$ depends on $S_{T(-n)\xi}(1)$, $n\in\mathbb N$, only and does not depend on the future ($S_{T(n)\xi}(1)$ for $n\geqslant 0$). Moreover, the impact of the past on the present attractor $\mathcal M_S(\xi)$ decays exponentially rapidly with respect to the time passed (in full accordance with our intuition). Note that this property is also violated on the level of pullback attractors. Uniform analogues of exponential attractors which are independent of time and where finite-dimensionality is replaced by an appropriate estimate for the Kolmogorov $\varepsilon$-entropy also appear in the literature (see [66], [234], and the references therein). Such constructions are usually based on straightforward generalizations of Theorem 6.17, so we give no further details here. An alternative possibility, where infinite-dimensional exponential attractors may appear, is the theory of dissipative PDEs in unbounded domains, where the global attractor is usually infinite-dimensional and we must use Kolmogorov entropy to control the size of the attractor (see [67], [179], and the references therein). Now we turn to the non-uniform (random) case where we have an ergodic measure $\mu$ for the underlying dynamical system $T(t)\colon\Psi\to\Psi$ and where the set $\mathcal B=\mathcal B(\xi)\in\Phi_1$, as well as the squeezing factor $L=L(\xi)$ in (6.14), are random variables. Namely, similarly to Theorem 6.10 we say that a discrete measurable cocycle $S_\xi(n)$ belongs to the class $\mathbb S_{\varepsilon,\kappa,L}(\mathcal B)$ if $S_\xi(1)\colon \mathcal O_\varepsilon(\mathcal B(\xi))\to \mathcal B(T(1)\xi)$, $\xi\in\Psi$, and $S_\xi(1)$ satisfies the squeezing property (6.14) for all $u_1,u_2\in\mathcal O_\varepsilon(\mathcal B(\xi))$. Then the following result holds. Theorem 8.19. Let $T(n)\colon\Psi\to\Psi$, $n\in\mathbb Z$, be a dynamical system on a Polish space $\Psi$ which possesses a Borel ergodic probability measure $\mu$. Let the cocycle $S_\xi(n)$ belong to the class $\mathbb S_{\varepsilon,\kappa,L}(\mathcal B)$ for some deterministic constants $\varepsilon>0$, $\kappa\in[0,1)$, and a random constant $L=L(\xi)$, let $\mathcal B(\xi)$ be a bounded random set in $\Phi_1$, and let the function $\xi\to \|\mathcal B(\xi)\|_{\Phi_1}$ be tempered. Also assume that (6.16) is satisfied and $\mathbb E(L^\theta)<\infty$. Then there exists a compact random set $\mathcal M_S(\xi)\subset \mathcal B(\xi)$ satisfying the following conditions. 1) The sets $\mathcal M_S(\xi)$ are semi-invariant: $S_\xi(n)\mathcal M_S(\xi)\subset \mathcal M_S(T(n)\xi)$ for all $n\in\mathbb N$ and almost all $\xi\in\Psi$, and their fractal dimensions are bounded for almost all $\xi\in\Psi$ by a deterministic constant. 2) The uniform attraction property
$$
\begin{equation}
\operatorname{dist}_{\Phi_1}\bigl(S_\xi(n)\mathcal B(\xi), \mathcal M_S(T(n)\xi)\bigr)\leqslant Ce^{-\beta n}, \qquad n\in\mathbb N,
\end{equation}
\tag{8.31}
$$
holds for almost every $\xi\in\Psi$ for some deterministic positive constants $C$ and $\beta$ which are independent of $\xi$ and $n$. 3) Let $\hat S_\xi(n)$ be another cocycle belonging to the class $\mathbb S_{\varepsilon,\kappa,L}(\mathcal B)$ such that for some deterministic $\delta$ and random $K(\xi)$ such that $\mathbb E(K^\theta)<\infty$. Then there exist a random variable $P(\xi)$ which is finite almost everywhere and is independent of $\delta$ and the concrete choice of $S_\xi$ and $\hat S_\xi$, and a deterministic constant $\gamma>0$ such thatThe proof of this theorem in the particular case $\kappa=0$ was given in [211]. The general case is completely analogous and so is omitted. Remark 8.20. We note that, similarly to the deterministic case, the random attractor obtained is not just a pullback exponential attractor, but also a forward attractor and the exponential rate of attraction is actually uniform. Moreover, we have a deterministic rate of attraction to it, which can be controlled in terms of the physical parameters of the system under consideration (as pointed out in the original paper [211], where exponential attractors for a random dynamical system were introduced). Some attempts to relax this uniform attraction rate and allow the constants $C$ and $\beta$ in (8.31) to be random were also made subsequently (see [247] and the references therein), but this did not lead to any essential simplifications of the assumptions on the cocycle $S_\xi(n)$ which could compensate for the drawbacks related to the loss of the uniform/deterministic attraction property. For instance, the random attractor $\mathcal A(\xi)$ constructed in Example 5.19 becomes “pullback exponential” if we allow the constant $C=C(\xi)$ in the exponential attraction property to be random, and then the control over the rate of attraction and Hölder continuity with respect to $\varepsilon\to0$ are lost. As in the deterministic case, passing from discrete to continuous exponential attractors requires some Hölder continuity in time and a random version of the transitivity of exponential attraction (see [211]). We finally mention that, in the estimate for Hölder continuity (8.33), we may only guarantee that the random variable $P(\xi)$ is finite almost everywhere. In order to verify that it has finite moments of some order, we need to control the rate of convergence in Birkhoff’s ergodic theorem. This problem is rather delicate and requires not only the finiteness of the moments of any order for $L(\xi)$ and $K(\xi)$, but also some kind of exponential mixing (in applications to stochastic equations we need the corresponding Ornstein–Uhlenbeck process to be exponentially mixing). We do not give any more details here and address the interested reader to [83], [99], [113], [142], [188], [211], and to the references therein. We conclude this subsection by a model example of a reaction-diffusion system perturbed by white noise. Example 8.21. Let $\Omega\subset\mathbb{R}^d$ be a bounded domain in $\mathbb{R}^d$ with smooth boundary. Consider the following reaction-diffusion system in $\Omega$: Following the standard scheme (see, for example, [57], [142], and the references therein), we associate with equation (8.34) a solution cocycle $S^\delta_\xi(t)\colon\Phi\to\Phi$ over the standard dynamical system on the canonical probability space $(\Psi,\mathcal F,\mu)$. Namely, analogously to Example 5.19, we consider the space $C_0(\mathbb{R})$ endowed with the locally compact topology, the scalar Wiener measure $\overline \mu$, and the dynamical system $(\theta(h)\overline\xi)(t):=\overline\xi(t+h)-\overline\xi(h)$. Then $\Psi$ is the space $[C_0(\mathbb{R})]^{\mathbb N}$ endowed with the Tychonoff topology, while $T(h)\colon\Psi\to\Psi$ and $\mu$ are the Cartesian products of the semigroups $\theta(h)$ and the measures $\overline\mu$, respectively. It is well known that the measure $\mu$ is ergodic (see, for example, [83]). The solution cocycle $S^\delta_\xi(t)\colon\Phi\to\Phi$ is constructed similarly to Example 5.19, by subtracting the corresponding Ornstein–Uhlenbeck process (see [211] for more details). Moreover, it is straightforward to verify (again, similarly to Example 5.19) that this cocycle possesses a tempered absorbing set $\mathcal B_\xi$ in $\Phi_1:=H^1_0(\Omega)$ with respect to the bornology of tempered random sets. In addition, this absorbing set is uniform with respect to the bornology of bounded deterministic subsets of $\Phi$. Thus, it is enough to construct the random exponential attractors $\mathcal M^\delta(\xi)$ for the sets $\mathcal B(\xi)$ only, so we may use the result of Theorem 8.19 to construct these attractors. We also note that the trajectories of the Wiener process are Hölder continuous in time for almost all $\xi\in\Psi$, so passing from discrete to continuous attractors does not cause any problems, and we only need to check the assumptions of Theorem 8.19 which was actually done in [211]. Thus, we have the following result, which was proved in [211]. Theorem 8.22. Let the assumptions stated above hold. Then the solution cocycle $S^\delta_\xi(t)\colon\Phi\to\Phi$, $\delta\in[0,1]$, associated with problem (8.34) possesses a random family $\mathcal M^\delta(\xi)$, $\delta\in[0,1]$, of exponential attractors which satisfy the following conditions. 1) They are semi-invariant and have finite fractal dimensions in $\Phi_1$. These dimensions are bounded by a deterministic constant independent of $\delta$ for almost all $\xi$. 2) They possess a uniform exponential attraction property for deterministic bounded sets of $\Phi$, that is, there is a positive constant $\alpha$ and a monotone function $Q$ such that for every bounded subset $B$ of $\Phi$ we have
$$
\begin{equation*}
\operatorname{dist}_{\Phi} \bigl(S_\xi^\delta(t)B,\mathcal M^\delta(T(t)\xi)\bigr) \leqslant Q(\|B\|_{\Phi})e^{-\alpha t}, \quad t\geqslant 0,
\end{equation*}
\notag
$$
for almost all $\xi$. 3) There exist an almost everywhere finite random variable $P(\xi)$ and a positive deterministic constant $\gamma$ which are independent of $\delta$ such that
$$
\begin{equation}
\operatorname{dist}_{\Phi}^{\rm sym} (\mathcal M^{\delta_1}(\xi),\mathcal M^{\delta_2}(\xi))\leqslant P(\xi)|\delta_1-\delta_2|^\beta.
\end{equation}
\tag{8.37}
$$
Moreover, in the case $\delta=0$ the corresponding exponential attractor $\mathcal M^0(\xi)$ is independent of $\xi$ and is the standard exponential attractor for the limit autonomous and deterministic reaction-diffusion system. Remark 8.23. We see that properly constructed exponential attractors remain robust with respect to white noise perturbations as well. This demonstrates an advantage of exponential attractors in comparison to global ones, where the limit deterministic attractor is not robust with respect to noise even in the ideal situation of regular attractors considered below: see Example 5.19 (see also [53] and [24] for the discussion related to stochastic bifurcation theory). We also mention that the conditions posed on the stochastic reaction-diffusion system (8.34) are far from being optimal and can be relaxed essentially, which is however beyond the scope of this survey. 8.2. Regular attractorsTo conclude this section, we consider briefly the class of dynamical systems whose global attractors have an exponential rate of attraction, so that they may also be considered as exponential ones. This usually happens when the dynamical system possesses a global Lyapunov function and all equilibria are hyperbolic. Then any complete trajectory on the attractor is a heteroclinic orbit between these equilibria, and the attractor is a finite collection of smooth finite-dimensional unstable submanifolds of equilibria. Following [10], such attractors are called regular and this is probably the only more or less general class of attractors where we are able to understand their structure; see [12], [26], [94], [95] for more details. In our exposition we mainly follow [228]. We restrict ourselves to considering only the case of discrete time (passing from discrete to continuous time is straightforward; see [228] for the details), so we assume that we are given a map $S\colon\Phi\to\Phi$ in a Banach space $\Phi$ and construct a discrete semigroup via $S(n):=S^n$ in $\Phi$. We make the following assumptions about the map $S$. Assumption A. The map $S$ belongs to $ C^1(\Phi,\Phi)$, and its Fréchet derivative $S'(u)$ is uniformly continuous on bounded subsets of $\Phi$; $S$ is injective and $\ker S'(u)=\{0\}$ for all $u\in\Phi$. Assumption B. The set of equilibria $\mathcal R_0$ of the map $S$ is finite: $\mathcal R_0:=\{u_1,\dots,u_N\}$ and every equilibrium is hyperbolic, the latter means that the spectrum of $S'(u_i)$ does not intersect with the unit circle for any $u_i\in\mathcal R_0$. Assumption C. The semigroup $S(n)\colon\Phi\to\Phi$ possesses a global attractor $\mathcal A$ in $\Phi$ endowed with the standard bornology of bounded subsets of $\Phi$. Assumption D. Any trajectory $u(n)=S(n)u_0$ stabilizes to one of the equilibria in $\mathcal R_0$, and there are no homoclinic structures, that is, if $u_1,\dots,u_k\in l^\infty(\mathbb Z)$ are complete bounded trajectories of $S(n)$ such that for some equilibria $v_1,\dots,v_{k+1}\in\mathcal R_0$, then all the $v_i$ are necessarily different. In applications Assumption D, the key one, usually follows from the existence of a global Lyapunov function. We recall that a continuous function $L\colon\Phi\to\mathbb{R}$ is called a global Lyapunov functional if the function $n\to L(S(n)u_0)$ is non-increasing along trajectories and the equality $L(S u_0)=L(u_0)$ implies that $u_0\in\mathcal R_0$. Together with Assumption C and the finiteness of the set $\mathcal R_0$, this implies not only the validity of Assumption D, but also the fact that any complete bounded trajectory is a heteroclinic orbit between two equilibria (see [12], [228] for more details). In turn, this gives the following description of the attractor $\mathcal A$: where $\mathcal M^+(u_i)$ is an unstable set of the equilibrium $u_i$, that is, the set of all initial data $u_0$ for which there exists a complete trajectory $u(n)$ converging to $u_i$ as $n\to-\infty$.We now use Assumption B on hyperbolicity to verify that the sets $\mathcal M^+(u_i)$ are actually finite-dimensional manifolds in $\Phi$ with dimensions equal to the instability index $\operatorname{ind}(u_i)$ which is the algebraic number of eigenvalues of $S'(u_0)$ lying outside the unit circle (it is easy to verify that a;; these numbers are finite due to the existence of a compact global attractor). The proof of this fact can be carried put similarly to our verification of the existence of inertial manifolds, so we do not present it here. Note only that first we construct the manifolds locally in a small neighbourhood of the equilibria and then extend them to global submanifolds using the assumption of injectivity (see Assumption A) and the absence of homoclinic structures. This gives us the fact that all $\mathcal M^+(u_i)$ are finite-dimensional submanifolds of $\Phi$ which are diffeomorphic to $\mathbb{R}^{\operatorname{ind}(u_i)}$ (see [228] for the details). Finally, we recall that, similarly to inertial manifolds, the manifolds $\mathcal M^+(u_i)$ possess an exponential tracking property, that is, any trajectory of $S(n)$ is attracted exponentially fast to some trajectory on $\mathcal M^+(u_i)$ until it occurs in a small neighbourhood of $u_i$. Moreover, due to Assumption D, any trajectory of $S(n)$ spends a finite time $T_\delta$ outside the $\delta$-neighbourhood of $\mathcal R_0$, and $T_\delta$ is uniform with respect to all trajectories starting from a bounded set. These two facts give us exponential attraction to $\mathcal A$ with the rate of convergence controlled by the number of equilibria and their hyperbolicity constants (see [228] for details). We summarize the results obtained in the following theorem. Theorem 8.24. Let the map $S\colon\Phi\to\Phi$ satisfy Assumptions A–D. Then the attractor $\mathcal A$ of the associated semigroup $S(n)\colon\Phi\to\Phi$ has the description (8.38), where $\mathcal M^+(u_i)$ are the $\operatorname{ind}(u_i)$-dimensional unstable submanifolds of the equilibria $u_i$. Moreover, any trajectory belonging to the attractor is a heteroclinic orbit between two equilibria, and the rate of attraction to $\mathcal A$ is exponential, that is, there exist a positive constant $\alpha$ and a monotone function $Q$ such that, for every bounded subset $B\subset\Phi$, Remark 8.25. Note that, despite the exponential attraction rate (8.39) for a regular attractor, it still makes sense to construct an exponential attractor even in the case where the global attractor is regular. The problem here is that the constant $\alpha$ and the function $Q$ are still not controllable in terms of the physical parameters of the dynamical system under consideration and we still may lose important intermediate dynamics (see Example 8.1). Note also that, although Assumption B is generic in a certain sense due to Sard’s theorem, the number of equilibria and their hyperbolicity constants can only be found/estimated explicitly in very exceptional cases, so even the attraction constant $\alpha$ in the exponential attraction rate is usually “unobservable”. We now turn to perturbation theory. To this end we assume that there is a family of cocycles $S_{\delta,\xi}(n)\colon\Phi\to\Phi$ for some dynamical system $T(n)\colon\Psi\to\Psi$ which depends on a parameter $\delta\in[0,1]$ (for simplicity we assume that the set $\Psi$ and the $T(n)$ are independent of $\delta$, although this is not essential). We assume that, for $\delta=0$, the corresponding maps $S_{\delta,\xi}(n)$ are independent of $\xi$ and $S_{0}(1)$ satisfies Assumptions A–D. Thus, in the limit, for $\delta=0$, we have a regular attractor. To specify the perturbation, we need two more assumptions. Assumption E. The maps $S_{\delta,\xi}(1)\in C^1(\Phi,\Phi)$ and the corresponding Fréchet derivatives $S'_{\delta,\xi}(1)$ are uniformly continuous on bounded subsets of $\Phi$ (also uniformly with respect to $\xi$ and $\delta$). Moreover, the maps $S_{\delta,\xi}(1)$ are injective for all $\xi$ and $\delta$ and $\ker{S'_{\delta,\xi}(1)}=\{0\}$ for all‘$\delta$, $\xi$ at any point of $\Phi$. Finally, we assume that where the constant $C$ depends on the norm $\|v\|_\Phi$ only (and is independent of $\xi$ and $\delta$). Assumption F. The family of cocycles $S_{\delta,\xi}(n)$ possesses a compact uniformly attracting set $\mathcal B\in\Phi$ with respect to the standard bornology of bounded sets in $\Phi$ (which is also uniform with respect to $\delta$). The perturbation theory of regular attractors is based on two relatively simple observations. 1) Since the cocycle $S_{\delta,\xi}(n)$ is close to the limit semigroup $S_0(n)$ and possesses a compact uniformly attracting set, the properties of perturbed trajectories will be close to the properties of non-perturbed ones. In particular, any perturbed trajectory will visit some $\varepsilon$-neighbourhood of the set of equilibria $\mathcal R_0$, where $\varepsilon=\varepsilon(\delta)$ tends to zero as $\delta\to0$. Moreover, due to the absence of homoclinic structures for the limiting dynamical system, this trajectory is unable to visit an appropriate neighbourhood of any $u_i\in\mathcal R_0$ more than once and spends a uniformly bounded time outside the $\varepsilon$-neighbourhood of $\mathcal R_0$ (see [228] for the details). This actually reduces the analysis to local perturbation theory near the hyperbolic equilibria. 2) Since the limit equilibria $u_i\in\mathcal R_0$ are hyperbolic, the saddle structure survives under small non-autonomous perturbations. In particular, any equilibrium $u_i\in\mathcal R_0$ generates an “equilibrium” $u_i(\xi)$ of the perturbed cocycle $S_{\delta,\xi}(n)$ if $\delta$ is small enough. This equilibrium is uniquely determined by the condition
$$
\begin{equation}
\sup_{\xi\in\Psi}\|u_i(\xi)-u_i\|_\Phi\leqslant C\delta.
\end{equation}
\tag{8.41}
$$
The “equilibria” $u_i(\xi)$ we have constructed remain hyperbolic, and the corresponding unstable manifolds $\mathcal M^+_\delta( u_i,\xi)$ are close to the unstable manifold $\mathcal M^+_0(u_i)$ of the limit problem as $\delta=0$ and possess the uniform exponential tracking property. In addition, due to the injectivity property these manifolds (which are initially defined for small neighbourhood of $u_i$ only) can be extended to global unstable submanifolds, which will be diffeomorphic to $\mathbb{R}^{\operatorname{ind}(u_i)}$. This allows us to establish an analogue of (8.38) for the non-autonomous (pullback) attractor $\mathcal A_\delta(\xi)$ for the perturbed cocycle $S_{\delta,\xi}(n)$:
$$
\begin{equation}
\mathcal A_\delta(\xi)=\bigcup_{i=1}^N\mathcal M_\delta^+(u_i,\xi).
\end{equation}
\tag{8.42}
$$
Moreover, we also have the property that any complete bounded trajectory is a heteroclinic orbit between two “equilibria” $u_i(\xi)$, as well as uniform exponential attraction of bounded sets. Finally, arguing as in the proof of Proposition 8.5, we establish uniform Hölder continuity of the following form:
$$
\begin{equation}
\operatorname{dist}_\Phi(\mathcal A_\delta(\xi),\mathcal A_0)\leqslant C\delta^\kappa
\end{equation}
\tag{8.43}
$$
for some positive $\kappa$ and $C$ which are independent of $\xi$ and $\delta$; see [228] for the details. We summarize the results obtained here in the following theorem. Theorem 8.26. Let the family of cocycles $S_{\delta,\xi}(n)\colon\Phi\to\Phi$ over a dynamical system $T(h)\colon\Psi\to\Psi$ depending on the parameter $\delta\in[0,1]$ satisfy Assumptions A–F. Then there exists $\delta_0>0$ such that, for every $\delta\leqslant\delta_0$, the non-autonomous (pullback) attractor $\mathcal A_\delta(\xi)$ of the cocycle $S_{\delta,\xi}(n)$ is a finite union of the unstable manifolds $\mathcal M^+_\delta(u_i,\xi)$ of the corresponding perturbed “equilibria” $u_i(\xi)$, $\xi\in\Psi$, of the limit hyperbolic equilibria $u_i\in\mathcal R_0$ (that is, (8.42) holds). Moreover, any complete bounded trajectory of the dynamical process $U_{\delta,\xi}(m,n)$ associated with the cocycle $S_{\delta,\xi}(n)$ is a heteroclinic orbit between $u_i(T(n)\xi)$ and $u_j(T(n)\xi)$ for some $i\ne j$. The rate of attraction to the attractors $\mathcal A_\delta(\xi)$ is uniform and exponential, that is, there exist a positive constant $\alpha$ and a monotone function $Q$ such that, for every bounded set $B\subset\Phi$ and every $m\geqslant n$, Remark 8.27. We see that the object obtained as a non-autonomous perturbation of a regular attractor gives us an example of a non-autonomous exponential attractor introduced before. In particular, it is not only a pullback, but also a forward in time exponential attractor. This gives us an evidence that the definition of a non-autonomous exponential attractor that we use is consistent and natural at least in the deterministic (uniform) case. Unfortunately, a reasonably general analogue of Theorem 8.26 does not seem to exist in the random (non-uniform) case; see Example 5.19. This is closely related to the well-known problem of developing the theory of centre (stable/unstable) manifolds for random/stochastic dynamical systems. The theory of random exponential attractors developed in [211] and discussed above may be one of the possible ways to handle this problem. 9. Determining functionalsIn this section we discuss an alternative approach to the justification of the finite-dimensional reduction in dissipative PDEs, which is related to the concept of determining functionals. This approach was introduced in [75] (see also [147]) for the case of the 2D Navier–Stokes equations and Fourier modes and was extended subsequently to many other classes of dissipative systems and various classes of determining functionals (see [41], [44], [46], [47], [77], [78], and the references therein). In our exposition we mainly follow the recent paper [114]. Definition 9.1. Let $S(t)\colon\Phi\to\Phi$ be a dynamical system acting on a Banach space $\Phi$. Then a finite system of continuous functionals $\mathcal F:=\{\mathcal F_1,\dots,\mathcal F_N\}$, $\mathcal F_i\colon\Phi\to\mathbb{R}$, is called asymptotically determining if for any two trajectories $u_1(t):=S(t)u_1$ and $u_2(t):=S(t)u_2$, the convergence implies that $\lim_{t\to\infty}\|u_1(t)-u_2(t)\|_{\Phi}=0$. Thus, if $\mathcal F$ is an asymptotically determining system, then the behaviour of any trajectory $u(t)$ as $t\to\infty$ is determined by the behaviour of finitely many quantities $\mathcal F_1(u(t)),\dots,\mathcal F_N(u(t))$. At first glance, this may look as a kind of finite-dimensional reduction, but a more detailed analysis shows that it is not the case in general since, in contrast to inertial forms constructed via Mané’s projection theorem or inertial manifolds, the quantities $\{\mathcal F_i(u(t))\}_{i=1}^N$ are not obliged to satisfy a system of ODEs. Actually, in many cases they satisfy a system of delay differential equations, but the phase spaces of such systems remain infinite-dimensional, so despite the widespread misunderstanding, determining functionals do not give any finite-dimensional reduction and are actually responsible for the reduction to a system of ODEs with delay. Nevertheless, such a reduction is also interesting from the theoretical point of view and has non-trivial applications to many related areas, for instance, to establishing the controllability of an originally infinite-dimensional system by finitely many modes (see, for example, [5]), to verifying the uniqueness of an invariant measure for random/stochastic PDEs (see, for example, [142]), and so on. We also mention more recent but promising applications of determining functionals to data assimilation problems where the values of the functionals $\mathcal F_i(u(t))$ are interpreted as the results of observations and the theory of determining functionals allows us to build new methods for recovering the trajectory $u(t)$ from the results of observations (see [5], [4], [184], and the references therein). In the case where the dynamical system under consideration possesses an attractor (with respect to the standard bornology of bounded sets in $\Phi$) we may use an alternative, non-equivalent but closely related concept of separating functionals. Definition 9.2. Let the dynamical system $S(t)\colon\Phi\to\Phi$ acting on a Banach space $\Phi$ possess an attractor $\mathcal A$ with respect to the standard bornology. Then a system $\mathcal F:=\{\mathcal F_i\}_{i=1}^N$ is separating on the attractor if for any two complete bounded trajectories $u_1(t)$ and $u_2(t)$ lying on the attractor the equality implies that $u_1(t)\equiv u_2(t)$. It is not difficult to show that (under the mild extra assumption that the $S(t)$ are continuous for every $t$), any separating system $\mathcal F$ of continuous functionals is automatically asymptotically determining (see, for example, [114]). The converse is not true in general: the corresponding counterexample which is on Example 2.7 is also given in [114]. This simple observation is rather useful since usually an attractor is more regular and the separation property is easier to verify. In order to illustrate the standard approach to determining functionals, we consider the model example of an abstract semilinear parabolic equation in a Hilbert space $\Phi$, where $A\colon D(A)\to\Phi$ is a self-adjoint linear positive operator with compact inverse and $F\colon\Phi\to\Phi$ is globally Lipschitz in $\Phi$ with Lipschitz constant $L$. Let $\{\lambda_i\}_{i=1}^\infty$ be the eigenvalues of the operator $A$ enumerated in the non-decreasing order and $\{e_i\}_{i=1}^\infty$ be the corresponding eigenvectors.Theorem 9.3. Let $N\in\mathbb N$ be such that $\lambda_{N+1}>L$. Then the system of functionals $\mathcal F_i(u)=(u,e_i)$, $i=1,\dots,N$ (which are the first $N$ Fourier modes) is asymptotically determining for the dynamical system generated by equation (9.1). Proof. Indeed, let $u_1(t)$ and $u_2(t)$ be two solutions of equation (9.1) such that
$$
\begin{equation*}
P_N(u_1(t)-u_2(t))\to0\quad\text{as}\quad t\to\infty,
\end{equation*}
\notag
$$
where $P_N$ is a spectral orthoprojector related to the first $N$ Fourier modes, and let $Q_N=1-P_N$. Let
$$
\begin{equation*}
v(t):=P_N(u_1(t)-u_2(t))\quad\text{and}\quad w(t):=Q_N(u_1(t)-u_2(t)).
\end{equation*}
\notag
$$
Then the last function solves the equation Multiplying this equation by $w$ and using that $F$ is globally Lipschitz, we end up with Using the assumption $\lambda_{N+1}>L$, we arrive at for some positive $\alpha$. Integrating this inequality and using that $\|v(t)\|_\Phi\to0$ as $t\to\infty$, we conclude that also $\|w(t)\|_{\Phi}\to0$, which finishes the proof of the theorem. Remark 9.4. The result of the last theorem can be generalized in a straightforward way to more general classes of linear determining functionals. For instance, if a system $\mathcal F:=\{\mathcal F_1,\dots,\mathcal F_N\}$ of linear continuous functionals satisfies the following inequality: We also note that, due to (9.3), the complete trajectory $u(t)$, $t\in\mathbb{R}$, is determined in a unique way by the values $\xi_i(t):=\mathcal F_i(u(t))$, that is,
$$
\begin{equation*}
u(t)=\mathfrak F(\xi_1(t+\cdot),\dots,\xi_N(t+\cdot)),\qquad \mathfrak F\colon [C(-\infty,0;\mathbb{R})]^N\to\Phi,
\end{equation*}
\notag
$$
and we have the exponential decay of the delay kernel: Thus, turning back to the quantities $\xi_n(t):=\mathcal F_n(u(t))$, we see that, in general, they do not satisfy a system of ODEs in $\mathbb{R}^N$, but satisfy a system of retarded ODEs with infinite delay and exponentially decaying delay kernel: Thus, since the phase space for such a retarded system remains infinite-dimensional, determining functionals do not provide any finite-dimensional reduction, but reduce the initial PDE to a system of ODEs with delay (which is often referred to as the Lyapunov–Schmidt reduction). We now give a more precise look at the number of elements in the “optimal” system of determining functionals. To this end, we give the following definition. Definition 9.5. Let $S(t)\colon\Phi\to\Phi$ be a dynamical system in a Banach space $\Phi$. Then the determining dimension $\dim_{\det}(S(t),\Phi)$ is the minimum number $N$ of continuous functionals $\mathcal F_1,\dots,\mathcal F_N\colon\Phi\to\mathbb{R}$ such that $\mathcal F:=\{\mathcal F_n\}_{n=1}^N$ is asymptotically determining. Note that we do not require the functionals $\mathcal F_n$ to be linear. Although in applications determining systems often consist of linear functionals, in the general theory the requirement of linearity looks artificial when a nonlinear dynamical system is considered. In addition, the use of nonlinear (for example, quadratic, cubic, and so on) functionals simplifies the theory and makes it more elegant. We recall that there exists a simple and natural lower bound for $\dim_{\det}(S(t),\Phi)$, which is related to the embedding dimension of the set $\mathcal R$ of equilibria points. Namely, by definition, $\dim_{\rm emb}(\mathcal R,\Phi)$ is the minimum number $M\in\mathbb N$ such that there exists a continuous injective map $\mathcal F\colon\mathcal R\to\mathbb{R}^M$. Then, obviously,
$$
\begin{equation}
\dim_{\det}(S(t),\Phi)\geqslant \dim_{\rm emb}(\mathcal R,\Phi).
\end{equation}
\tag{9.4}
$$
Indeed, any asymptotically determining system must, in particular, distinguish different equilibria. Note also that $\dim_{\rm emb}(\mathcal R,\Phi)$ is a topological invariant and can be estimated using, for example, Lebesgue covering dimension. Another interesting observation is that, due to Sard’s theorem, the set of equilibria $\mathcal R$ is generically finite (see [12], [193]); in particular, it is so for equation (9.1). Therefore, in a generic situation, (9.4) gives us $\dim_{\det}(S(t),\Phi)\geqslant 1$ for the lower bound. One of the most surprising results of the theory is that this estimate is sharp, that is, there exists a dense set of smooth functionals such that each of them is asymptotically determining. Theorem 9.6. Let the set $\mathcal R$ of equilibria be finite. Then the determining dimension of the solution semigroup $S(t)\colon\Phi\to\Phi$ associated with equation (9.1) is equal to one. Moreover, there is a dense/prevalent set of polynomial functionals on $\Phi$ such that each of them is a determining functional for $S(t)$. Idea of the proof. The proof is based on the Hölder continuous infinite-dimensional version of the famous Takens delay embedding theorem (see [193], [203], [213], and the references therein). To apply this theorem, we need to control the size of the set $\mathcal R$ of equilibria (which can be done by our assumptions) and the size of the sets of periodic orbits of period $\tau,2\tau,\dots,k\tau$, where $k$ is large enough. This control is attained using the fact that if $\tau\ll1$, then all such periodic orbits must be equilibria (see [193] for the details). On the other hand, the fractal dimension of the attractor $\mathcal A$ is finite and we have the one-to-one Hölder-continuous Mané projection of this attractor onto the finite-dimensional set $\overline{\mathcal A}\subset\mathbb{R}^N$ with the projected dynamical system $\overline S(t)$ on it.
Thus, by an appropriate version of Takens’s delay embedding theorem (see, for example, [193]) there exists a prevalent set of continuous functionals $\mathcal F\colon\Phi\to\mathbb{R}$ (polynomials of sufficiently large degree are enough) such that the map
$$
\begin{equation*}
\mathbb F_k\colon\mathcal A\to \mathbb{R}^k,\qquad \mathbb F_k(u_0):=\bigl(\mathcal F(u_0),\mathcal F(S(\tau)u_0),\dots, \mathcal F(S((k-1)\tau)u_0)\bigr),
\end{equation*}
\notag
$$
is one-to-one on the attractor $\mathcal A$ if $k$ is large enough and $\tau>0$ is small enough. Thus, each of these functionals is separating trajectories on the attractor and therefore is asymptotically determining (see [114], [193] for more details). Remark 9.7. The result of Theorem 9.6 also gives a reduction to a delayed ODE. Namely, as shown in [114], there exists a continuous function $\Theta\colon\mathbb{R}^k\to \mathbb{R}$ such that the observable $\xi(t):=\mathcal F(u(t))$, where $u(t)$ is any complete trajectory on the attractor, solves the scalar ODE with finite delay: and if the values of $\xi(t-\tau),\xi(t-2\tau),\dots,\xi(t-k\tau)$ are known for some $t\in\mathbb{R}$, then the corresponding point $u(t)\in\mathcal A$ can be recovered using the attractor reconstruction procedure provided by Takens’ delay embedding theorem. The situation when the set $\mathcal R$ is not-finite looks similar. Proposition 9.8. Let $S(t)\colon\Phi\to\Phi$ be a dynamical system associated with equation (9.1), and let $\mathcal R$ be its set of equilibria. Then the determining dimension of $S(t)$ satisfies The proof of this proposition is analogous to the proof of Theorem 9.6 and is presented in [114]. We expect that the left-hand inequality (9.6) is actually equality, but we have not checked this so far. Moreover, an analogue of delay ODE (9.5) also holds in this case (see [114]). Remark 9.9. We see that the determining functionals are indeed responsible for a reduction of the original PDE to ODEs with delay and are not related to any kind of finite-dimensional reduction. In addition, the minimum number of determining functionals (determining dimension) is related to the “size” of the set of equilibria $\mathcal R$ of the dynamical system under consideration only (and is not related to any dynamical properties of this dynamical system). Although these results are stated for a semilinear parabolic system (9.1) only, their close analogues remain true for much wider class of equations including the Navier–Stokes system, damped wave equations, and so on. We conclude this section by several illustrative examples. We start with the best studied case of one spatial dimension. Example 9.10. Consider the following 1D semilinear heat equation: Moreover, the equilibria $\mathcal R$ of this problem satisfy the second-order ODE Thus, the map $u\to u'(0)$ provides a homeomorphic (and even smooth) embedding of $\mathcal R$ in $\mathbb{R}$, so $\dim_{\rm emb}(\mathcal R,\Phi)=1$. Thus, we expect that $\dim_{\det}(S(t),\Phi)=1$ (or at most $2$ in accordance with Proposition 9.8). The possible explicit form of the determining functional is well known here: $\mathcal F(u):= u\big|_{x=x_0}$, where $x_0>0$ is small enough (see [140]). Indeed, let $u_1(t),u_2(t)\in\mathcal A$ be two complete trajectories of (9.7) lying on the attractor such that $u_1(t,x_0)\equiv u_2(t,x_0)$. Then the function $v(t)=u_1(t)-u_2(t)$ solves the equation
$$
\begin{equation}
\partial_t v=\nu\,\partial_x^2v-l(t)v,\quad v\big|_{x=0}=v\big|_{x=x_0}=0,
\end{equation}
\tag{9.8}
$$
where Since the attractor $\mathcal A$ is bounded in $C[0,\pi]$, we know that $l(t)$ is globally bounded in $C[0,\pi]$, that is, $|l(t)|_{C[0,\pi]}\leqslant L$. Multiplying equation (9.8) by $v$, integrating with respect to $x\in[0,x_0]$, and using the fact that the first eigenvalue of $-\partial_x^2$ with Dirichlet boundary conditions is $(\pi/x_0)^2$, we get
$$
\begin{equation*}
\frac{1}{2}\,\frac{d}{dt}\|v(t)\|^2_{L^2}+\nu\biggl(\frac\pi{x_0}\biggr)^2 \|v(t)\|^2_{L^2}-L\|v(t)\|^2_{L^2}\leqslant0.
\end{equation*}
\notag
$$
Fixing $x_0>0$ to be small enough so that $\nu(\pi/x_0)^2>L$, applying Gronwall’s inequality, and using that $\|v(t)\|_{L^2}$ remains bounded as $t\to-\infty$, we conclude that $v(t)\equiv0$ for all $t\in\mathbb{R}$ and $x\in[0,x_0]$. Thus, the trajectories $u_1(t,x)$ and $u_2(t,x)$ coincide for all $t$ and all $x\in[0,x_0]$. Using now arguments related to logarithmic convexity or Carleman-type estimates, which work for much more general class of equations (see [200]), we conclude that the trajectories $u_1$ and $u_2$ coincide. Thus, $\mathcal F$ is separating on the attractor and therefore is asymptotically determining. Being pedantic, we need to note that the functional $\mathcal F(u)=u\big|_{x=x_0}$ is not defined on the phase space $H$, but rather on the proper subspace $C[0,\pi]$ of it (this is a typical situation for determining nodes; see [184]). However, we have an instantaneous $\Phi\to C[0,\pi]$ smoothing property, thus if we start from $u_0\in \Phi$, then the value $\mathcal F(u(t))$ will be defined for all $t>0$, so we just ignore this small inconsistency. Example 9.11. Let us consider the same equation (9.7) on $[0,\pi]$, but endowed with periodic boundary conditions. In this case we do not have the condition $v(0)=0$, so the set $\mathcal R$ of equilibria is naturally embedded in $\mathbb{R}^2$, rather than in $\mathbb{R}^1$, by the map $u\to \bigl(u\big|_{x=0},u'\big|_{x=0}\bigr)$. Thus, we cannot expect that the determining dimension is 1. Moreover, at least in the case where $g=\operatorname{const}$ equation (9.7) possesses a spatial shift as a symmetry, and therefore each non-trivial equilibrium generates a whole circle of equilibria. Since a circle cannot be homeomorphically embedded in $\mathbb{R}^1$, the determining dimension must be at least $2$. We claim that it is indeed $2$ and determining functionals can be taken in the form Indeed, arguing exactly as in the previous example, we see that the system $\mathcal F=\{\mathcal F_1,\mathcal F_2\}$ is asymptotically determining if $x_0>0$ is small enough. The next natural example shows that the determining dimension may be finite and small even when the corresponding global attractor is infinite-dimensional. Example 9.12. Consider the semilinear heat equation (9.7) on the whole line $x\in\mathbb{R}$. The natural phase space for this problem is the uniformly-local space Nevertheless, the system of linear functionals (9.9) remains determining for this equation by exactly the same reasons as in Examples 9.10 and 9.11. Thus, One functional is not determining in general for the reasons explained in Example 9.11.We now give an example of a non-dissipative and even conservative system with determining dimension 1. Example 9.13. Consider the following 1D wave equation: Remark 9.14. The principal difference between the case of one spatial dimension and the multi-dimensional case is that in the 1D case any equilibrium $u_0\in\mathcal R$ of such a PDE solves a system of ODEs, so the dimension of $\mathcal R$ is restricted by the order of this system. This allows us in many cases to get sharp estimates for the determining dimension and even compute it explicitly. In particular, Examples 9.10 and 9.11 clearly show that it is independent of the physical parameters of the dynamical system, as well as of the dimension of the attractor. In contrast to this, in the multi-dimensional case $u_0\in\mathcal R$ usually solves an elliptic PDE and we do not have any good formulae for the dimension of $\mathcal R$. The best we can do in general is to use the obvious estimate 10. Appendix. Function spacesThe aim of this appendix is to introduce and discuss various classes of function spaces which are used throughout the survey. We start with Lebesgue and Sobolev spaces. Definition 10.1. Let $\Omega$ be a domain in $\mathbb{R}^d$ with a sufficiently smooth boundary. We denote by $L^p(\Omega)$, $1\leqslant p\leqslant\infty$, the Lebesgue space of functions whose $p$th power is Lebesgue integrable. For any $n\in\mathbb N$, we denote by $W^{n,p}(\Omega)$ the Sobolev space of distributions whose derivatives up to order $n$ inclusive belong to the space $L^p(\Omega)$. For non-integer positive $s=n+\alpha$, $n\in\mathbb Z_+$, $\alpha\in(0,1)$, the space $W^{s,p}(\Omega)$ is defined as a Besov space $B^s_{p,p}(\Omega)$ via the following norm: Now let $V$ be a Banach (or more generally, a locally convex) space, and let $V^*$ be its dual space (the space of linear continuous functionals on $V$). Then the weak topology on $V$ is defined by the following system of seminorms on $V$: $p_l(x):=|lx|$, $l\in V^*$. On the level of sequences this means that $x_n\rightharpoondown x$ in $V$ if and only if $lx_n\to lx$ for all $l\in V^*$. The weak-star topology on the dual space $V^*$ is defined analogously (see, for example, [191], [201], and the references therein). The key result, which is widely used in the theory of attractors, is the Banach–Alaoglu theorem, which claims that the closed unit ball in $V^*$ is compact in the weak-star topology (this result can be extended to locally convex or even linear topological spaces if we replace the unit ball by the polar $U^0\in V^*$ of any bounded set $U\subset V$; see [191] for the details). We recall that a set $U$ is bounded in a linear topological space if it is absorbed by any neighbourhood of zero, and the polar $U^0$ is defined by
$$
\begin{equation*}
U^0:=\Bigl\{l\in V^*\colon\sup_{x\in U} |lx|\leqslant 1\Bigr\}.
\end{equation*}
\notag
$$
The analogous results hold for the weak topology if and only if the space $V$ is reflexive ($V=V^{**}$). Most important for us is the fact that the Sobolev spaces $W^{s,p}(\Omega)$ are reflexive if and only if $1<p<\infty$ (see [217]). We also recall that compactness and sequential compactness are different (unrelated) concepts in non-metrizable topological spaces, so the Banach–Alaoglu theorem (which is based on Tychonoff’s compactness theorem) does not give a sequential weak-star compactness of the unit ball in the dual space. In order to get sequential compactness we need to assume that the space $V$ is either separable (then the unit ball in the dual space is metrizable in the weak-star topology) or reflexive (then weak compactness and weak sequential compactness coincide due to the Eberlein–Smulian theory); see [191] for more details. We also recall that $L^\infty(\Omega)$ is the dual space of $L^1(\Omega)$, so there is a natural choice of a weak-star topology in $L^\infty(\Omega)$ which gives the compactness and sequential compactness of a unit ball. In contrast to this, the spaces $C(\overline\Omega)$ and $L^1(\Omega)$ are not dual to any Banach spaces, and there is no reasonable topology in them that gives the compactness of their unit balls. In order to consider evolutionary equations, we often use the spaces of functions $u(t)$ with values in some Banach space $V$. Here usually $t$ ranging over $\mathbb{R}$ or a subset of $\mathbb{R}$ is interpreted as time and $V$ is an appropriately chosen Sobolev space. For any $[a,b]\subset\mathbb{R}$ and any $1\leqslant p\leqslant\infty$ we define $L^p(a,b;V)$ as the space of Bochner measurable functions such that
$$
\begin{equation*}
\|u\|_{L^p(a,b;V)}^p:=\int_a^b\|u(t)\|_V^p\,dt<\infty.
\end{equation*}
\notag
$$
The space of functions that are Bochner measurable and locally integrable to power $p$ will be denoted by $L^p_{\rm loc}(a,b;V)$. Sometimes, when we need to emphasize the properties of functions near the finite endpoints $a$ and $b$, we will write, say, $L^p_{\rm loc}([a,b],V)$ or $L^p_{\rm loc}((a,b),V)$. The time Sobolev and Besov spaces $W^{s,p}(a,b;V)$ are defined analogously to (10.1); see [150] and the references therein. The uniformly local space $L^p_b(a,b;V)$ is defined as the subset of $L^p_{\rm loc}(a,b;V)$ for which the following norm is finite:
$$
\begin{equation*}
\|u\|_{L^p_b(a,b;V)}:=\sup_{s\in\mathbb{R},\,[s,s+1]\subset [a,b]}\|u\|_{L^p(s,s+1;V)}
\end{equation*}
\notag
$$
and in the case $b-a<1$ we just set $\|u\|_{L^p_b(a,b;V)}:=\|u\|_{L^p(a,b;V)}$. The spaces $W^{s,p}_b(a,b;V)$ are defined analogously. We also need some other properties of the locally convex (Fréchet) space $L^p_{\rm loc}(\mathbb{R},V)$. We assume that the space $V$ is reflexive and separable. Then $[L^p(a,b;V)]^*=L^q(a,b;V^*)$ for $1/p+1/q=1$, $1\leqslant p<\infty$. In particular, these spaces are reflexive and separable for any $p$, $1<p<\infty$, and therefore the Banach–Alaoglu theorem gives us the weak sequential compactness of the unit ball in $L^p(a,b;V)$. This, in turn, gives us the weak sequential compactness of any bounded weakly closed subset of $L^p_{\rm loc}(\mathbb{R},V)$. In particular, if this bounded set is convex, then closedness in the strong topology is enough. For instance, the unit ball in $L^p_b(\mathbb{R},V)$ is weakly sequentially compact in $L^p_{\rm loc}(\mathbb{R},V)$. We use this fact, in particular, to define the hulls of translation bounded external forces. Recall that $L^p_{\rm loc}(\mathbb{R},V)$ with the weak topology can be obtained as the projective limit of the spaces $L^p(-n,n;V)$ (endowed with the weak topology) as $n\to\infty$. In particular, $u_n\rightharpoondown u$ in $L^p_{\rm loc}(\mathbb{R},V)$ if and only if for any $n\in\mathbb N$ and any $l\in L^q(-n,n;V^*)$ we have the convergence
$$
\begin{equation*}
\int_{-n}^n\langle l(t),u_n(t)\rangle\,dt\to \int_{-n}^n\langle l(t),u(t)\rangle\rangle\,dt.
\end{equation*}
\notag
$$
The author thanks V. Chepyzhov, A. Ilyin, V. Kalantarov, A. Kostianko and D. Turaev for many stimulating discussions.
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