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On weak solvability of fractional models of viscoelastic high order fluid V. G. Zvyagin, V. P. Orlov Voronezh State University
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     § 1. IntroductionConsider the motion of an incompressible fluid occupying a domain $\Omega\subset \mathbb{R}^N$ in $\mathbb{R}^N$, $N=2,3$, on the time interval $[0,T]$, $T>0$. We will assume that $\Omega$ is a bounded domain with locally Lipschitz boundary $\partial\Omega$, $Q_T=[0, T]\times \Omega$. Let $v(t,x)=(v_1,\dots,v_N)$ be the velocity vector at a point $x\in \Omega$ at time $t$. The Cauchy motion equation has the form (see [1])
$$
\begin{equation}
\rho\biggl(\frac{\partial v}{\partial t} +\sum_{i=1}^Nv_i\,\frac{\partial v}{\partial x_i}\biggr) +\nabla p-\operatorname{Div}\sigma=\rho f,\qquad (t,x)\in Q_T,
\end{equation}
\tag{1.1}
$$
where $\rho=\mathrm{const}>0$ is the fluid density (which is assumed to be 1), $p=p(t,x)$ is the pressure at $x$ at time $t$, $\sigma$ is the deviator of the stress tensor, and $f$ is the density of external forces. By $\operatorname{Div}$, we denote the divergence of a matrix-function, that is, $\operatorname{Div}\sigma$ is the vector whose entries are the divergences of the rows of a matrix $\sigma$. Equation (1.1) is augmented with the incompressibility condition $\operatorname{div}v=0$ and a rheological relation determining the type of the fluid. The rheological relation
$$
\begin{equation}
\biggl(1+\sum_{k=1}^Lp_kD_t^{a_k}\biggr)\sigma = \nu\biggl(1+\nu^{-1}\sum_{k=1}^{M}q_kD_t^{b_k}\biggr)\mathcal{E}(v),
\end{equation}
\tag{1.2}
$$
relates the deviator of the stress tensor $\sigma$ and the strain rate tensor $\mathcal{E}(v)$ defines an extensive class of viscoelastic continuous media. Here, $L$, $M$ are natural numbers, $p_k,q_k\geqslant 0$, $a_k, b_k\in [k,k+1)$, $a_L,b_M> 0$, $\nu>0$, and $D_t^r$ is the fractional Riemann–Liouville derivative of order $r$. The strain rate tensor $\mathcal{E}(v)$ is defined as the matrix $\mathcal{E}(u)$=$\{\mathcal{E}_{ij}(u)\}_{i,j=1}^N$ with entries $\mathcal{E}_{ij}(u)=\frac12(\partial u_i/\partial x_j +\partial u_j/\partial x_i )$. In the case of integer $a_k$, $b_k$, in (1.2) $a_k=k$, $b_k=k$, and $D_t^r=d^r/dt^r$ are ordinary derivatives. Hence (see [2]) for $M=L\,{-}\,1$, $L=1, 2, \dots$, the medium is called a Maxwell fluid of order $L$; for $M=L$, $L=1, 2, \dots$, the medium is called an Oldroyd fluid of order $L$; for $M=L+1$, $L=0, 1, \dots$, the medium is called a Pavlovsky fluid of order of $L$. Integer motion models of viscoelastic media of order $b\leqslant 2$, $b=\max(a_L,b_M)$, have been studied quite well. Equation (1.2) in this case defines many well-known models (see, for example, [3], [4] and the references there). In particular, the Newton, Maxwell, Voigt, and Oldroyd models have the order $b=1$. The Zener, anti-Zener, and Burgers models have the order $b=2$. For a mechanical interpretation of standard models and a detailed bibliographic survey, see [3]. Motion models of viscoelastic media of order $b= 2$ has been extensively studied (see, for example, [5]–[8]). Various settings of initial boundary value problems were considered and solvability and properties of their solutions were investigated. The motion equations of linear viscoelastic fluids with finite number of discretely distributed relaxation and lag times obeying the defining equation (1.2) of high orders (see [2], [9]) can be obtained either in the form of differential equations, or in the form of integro-differential ones. Exclusion of $\sigma$ from equation (1.2) by differentiating (1.1) with respect to $t$ and selecting a suitable linear combination of the differentiation results using (1.2) leads to an initial boundary value problem with respect to unknown functions $v$ and $p$. In this case, the system of equations contains high-order derivatives of unknown functions with respect to $t$ and $x$. The existence and uniqueness of generalized solutions in classes of sufficiently smooth functions with sufficiently smooth data for the initial boundary value problems thus obtained were established in [10]–[13]. Another type of initial boundary problems corresponding to the high-order equation (1.2) is obtained by integrating equation (1.2) and finding $\sigma$ in the form
$$
\begin{equation}
\sigma(t,x)=\mu_0\mathcal{E}(t,x)+\int_0^tG(t-s)\mathcal{E}(v)(s,x)\,ds+f_*,
\end{equation}
\tag{1.3}
$$
or
$$
\begin{equation}
\sigma(t,x)=\int_0^tG(t-s)\mathcal{E}(v)(s,x)\,ds+f_*
\end{equation}
\tag{1.4}
$$
followed by a substitution of $\sigma$ in (1.1) (see [2]). Here, $f_*$ contains the initial values of $\sigma$ and $v$ and of their derivatives. Existence theorems similar to those for Navier–Stokes systems were established for the corresponding integro-differential problems (see, for example, [14], [15]). Equations (1.3) and (1.4) show (see [4], [16]) that the corresponding fluids are media with long-term memory with respect to spatial variables $x$, since the state of the deviator of the stress tensor $\sigma (t,x)$ at time $t$ depends on the values of the strain rate tensor $\mathcal{E}(v)(s,x)$ for all $s\in [0, t]$. Of great interest (as more realistic from different points of view) are models that take into account the state of the medium along the trajectories of the velocity field, which are defined as solutions to the Cauchy problem
$$
\begin{equation}
z(\tau; t, x)=x+\int_t^\tau v(s, z(s; t, x))\,ds, \qquad 0\leqslant t, \tau\leqslant T, \quad x\in\overline{\Omega}
\end{equation}
\tag{1.5}
$$
(see, for example, [17]). Weak solvability of initial boundary value problems for Oldroyd, Voigt, anti-Zener, and Newton models with memory along trajectories was studied in [18]–[24]. Note that in this case, finding trajectories requires the solution of the Cauchy problem for a system of ODE defined by the velocity field $v$. In the case of weak solvability, $v$ belongs to a Sobolev space, and since the classical solution is not guaranteed to exist, the solvability of the Cauchy problem should be considered in the class of regular Lagrangian flows (RLFs), which generalize the concept of a classical solution of ODE systems (see § 4 below). Note that the replacement of the usual derivative $D_t$ by the substantial one in the integer relation (1.2) implies the presence of velocity field trajectories in integro-differential summands in the motion equation (see [19]). However, integer models are not capable of adequately describing the behaviour of many polymers and complex biophysical fluids. The transition to models with fractional derivatives in the rheological relation stems form the necessity of dealing with large class of materials in which it is necessary to take into account the effects of creep, relaxation and elasticity (see [3], [25]–[29]). Fractional models provide a more generalized form of conventional viscoelastic models and relate more closely to the actual physical and mechanical properties of polymers and physiological fluids. It was found that the application of concepts and methods of the theory of fractional derivatives calculus provides an adequate tool for modeling the viscoelastic behaviour of mechanical dynamic systems with memory, a number of physical phenomena, biological processes, and hereditary elastic deformation (see [5]–[8]). Weak solvability and properties of solutions for fractional models with memory of order $b\leqslant 2$ were studied in [30]–[33]. Below we investigate the weak solvability of high-order fractional Oldroyd models with memory along the trajectories of the velocity field, the RLF properties being used substantially. The result of the present paper was announced in [34]. The paper is organized as follows. Notation and auxiliary facts are given in § 2. The initial boundary value problem for the high-order Oldroyd model is posed in § 3. Necessary information on regular Lagrangian flows is given in § 4. The weak solvability theorem for the initial boundary problem for the Oldroyd model, which is the main result of the present paper, is formulated in § 5. In § 6 auxiliary regularized problems are studied. The proof of the main result is given in § 7. Proposition 1 from § 3 is given in § 8. The constants in inequalities and chains of inequalities that do not depend on essential parameters are denoted by a single letter $M$ (with indexes, if necessary). § 2. Notation and auxiliary resultsWe will need the function spaces $V$ and $H$ (see p. 20 in [35]) of divergence free functions. The space $V=\{v\colon v\in \mathring{W}_2^1(\Omega)^N$, $\operatorname{div} v=0\}$ is Hilbert space with inner product $(v,u)_V=\sum_{i,j=1}^N\int_\Omega\mathcal{E}_{ij}(u)\cdot\mathcal{E}_{ij}(v)\,dx$ and the corresponding norm. By the Korn inequality $\sum_{i,j=1}^N\int_\Omega\mathcal{E}_{ij}(v)\cdot\mathcal{E}_{ij}(v)\,dx +\|v\|^2_{L_2(\Omega)^N}\geqslant c \int_\Omega\sum_{i,j=1}^N|\partial v_i/\partial x_j|^2\,dx$, $c>0$ (see [36]), the norm on $V$ is equivalent to the norm $\|v\|_{W_2^1(\Omega)^N}=\|v\|_{L_2(\Omega)^N} +\sum_{i,j=1}^N\|\partial v_i/\partial x_j \|_{L_2(\Omega)^N}$ induced from the space $ W_2^1(\Omega)^N$, The space $H$ is the closure of $V$ in the $L_2(\Omega)^N$-norm, and $V^{-1}$ is the dual space to $V$. The value of a functional $g\in V^{-1}$ evaluated on $u\in V$ is denoted by $\langle g,u\rangle$. The norms in the spaces $H$ and $V$ will be denoted by $|\,{\cdot}\,|_0$ and $|\,{\cdot}\,|_1$, respectively, the norm in $L_2(0,T;V)$ is denoted by $\|\,{\cdot}\,\|_{0,1}$, and $\|\,{\cdot}\,\|_{0,-1}$ is the norm on the space $L_2(0,T;V^{-1})$. The inner product in the Hilbert spaces $L_2(\Omega)$, $H$, $V$, $L_2(\Omega)^N$, $L_2(\Omega)^{N\times N}$ is denoted by $(\,{\cdot}\,,{\cdot}\,)$ (the corresponding spaces should be clear from the context). § 3. Statement of the initial boundary value problemLet $M=L$ in the rheological equation (1.2), and let $m$ be their common value, so that $M=L=m$. Let the higher derivatives orders $a_m$ and $b_m$ in (1.2) be equal, $b_m=a_m$, $a_m\in (m,m+1)$, and let the coefficients at higher derivatives $p_m, q_m$ be positive. Then the rheological equation (1.2) defining the Oldroyd model takes the form
$$
\begin{equation}
\biggl(1+p_mD_t^{a_m}+\sum_{k=1}^{m-1}p_kD_t^{a_k}\biggr)\sigma =\nu\biggl(1+q_mD_t^{a_m}+\nu^{-1}\sum_{k=1}^{m-1}q_kD_t^{b_k}\biggr) \mathcal{E}(v), \qquad \nu>0.
\end{equation}
\tag{3.1}
$$
Here, $m$ is a natural number, $a_k,b_k\in [k,k+1)$ for $k=1,\dots, m-1$. Let us recall some facts on fractional derivatives and integrals (see, for example, [3] and [37], § 1.2). The fractional Riemann–Liouville integrals $I_{0t}^{\alpha}$ and the derivatives $D_{0t}^{\alpha}$ of positive order $\alpha$ of a function $y(t)$ on $[0,T]$ are given by
$$
\begin{equation}
\begin{aligned} \, I_t^{\alpha}y &= \frac{1}{\Gamma(\alpha)} \int_0^t(t-s)^{\alpha-1}\,y(s)\,ds, \qquad \alpha>0, \\ D_t^{\alpha}y &= \frac{1}{\Gamma(n-\alpha)}\, \frac{d^n}{dt^n} \int_0^t(t-s)^{n-1-\alpha}\,y(s)\,ds, \qquad n= [\alpha]+1, \end{aligned}
\end{equation}
\tag{3.2}
$$
where $\Gamma(\alpha)$ is the Euler gamma function. In particular, for $0<\alpha<1$,
$$
\begin{equation*}
D_{0t}^{\alpha}y(t)=\frac{1}{\Gamma(1-\alpha)}\, \frac{d}{dt}\, \int_0^t(t-s)^{-\alpha}\,y(s)\,ds,
\end{equation*}
\notag
$$
and, for any natural number $n$, which is the usual derivative of order $n$. The fractional differentiation operator $D_t^{\alpha}$ is the left inverse of the fractional integration operator: $D_t^{\alpha}I_t^{\alpha}y(t)=y(t)$, and for $y(t)=I_t^{\alpha}z(t)$, $z(t)\in L_1(0,T)$, we have $I_t^{\alpha}D_t^{\alpha}y(t)=y(t)$. The semigroup property $I_t^{\alpha}I_t^{\beta}=I_t^{\alpha+\beta}$, $\alpha,\beta>0$ holds. For $\alpha\leqslant \beta$, we have $D_t^{\alpha}I_t^{\beta}y(t)=I_t^{\beta-\alpha}y(t)$, and, for $y(t)=I_t^{\beta}z(t)$, $z(t)\in L_1(0,T)$, we have $I_t^{\alpha}D_t^{\beta}y(t)=I_t^{\beta-\alpha}y(t)$ for $\alpha\geqslant \beta$. Let us express $\sigma$ from formula (3.1) via $\mathcal{E}(v)$. Let us use the relation (see p. 50 in [37])
$$
\begin{equation}
I_t^aD_t^{a}y(t)=y(t)+ \sum_{k=1}^n\frac{y_{n-a}(0)t^{a-k}}{\Gamma (a-k+1)},\qquad y_{n-a}(t)=I_t^{n-a}y(t).
\end{equation}
\tag{3.3}
$$
Let
$$
\begin{equation*}
\mathcal{I}(\alpha)=\{y: y=I_t^{a}w,\ w\in (L_1(0,T;L_2(\Omega)^{N\times N})\}.
\end{equation*}
\notag
$$
If $y\in \mathcal{I}(\alpha)$, then the second term in (3.3) is zero (see p. 49 in [37]). Proposition 1. For all $\sigma, \mathcal{E}(v) \in \mathcal{I}(\alpha)$, where $G(t-s)$ satisfies the estimate $|G(t-s)|\leqslant M(t-s)^{\gamma_1-1}$, $\gamma_1=\min (a_m-b_{m-1}, a_m-a_{m-1})$, $\mu_0=p_m^{-1}q_m$. It is obvious that $0<\gamma_1<2$. The proof of Proposition 1 is given in § 8. The assumption $\sigma, \mathcal{E}(v) \in \mathcal{I}(\alpha)$ shows that we do not take into account the relationships between the initial values $\sigma(0,x)$, $\mathcal{E}(v)(0,x)$ and their derivatives ($f_*$ in (1.3) and (1.4)) and nullify them. Otherwise, $f_*$ can be plugged into the right-hand side of $f$ in (1.1) (we shall not dwell on this point). The defining equation (3.4) shows that Oldroyd fluids are media with long-term memory on spatial variables in the sense that the state of the stress tensor deviator $x\in \Omega$ at any time $t$ and each $x\in \Omega$ depends not only on the state $\mathcal{E}(v)(t,x)$ at time $t$, but also depends on the values of $\mathcal{E}(v)(s,x)$ for all $s\in [0, t]$ at the same point $x\in \Omega$ (see [4]). Substituting (3.4) into (1.1) (in what follows, we assume $\rho=1$ for simplicity), we get the equation
$$
\begin{equation}
\frac{\partial v}{\partial t}+\sum_{i=1}^nv_i\, \frac{\partial v}{\partial x_i} -\mu_0\Delta v -\operatorname{Div}\int_0^t G(t-s)\mathcal{E}(v)(s,x)\,ds + \nabla p = f, \qquad (t,x)\in Q_T.
\end{equation}
\tag{3.5}
$$
By taking into account the memory along the trajectories $z(\tau,t,x)$ of fluid motions, which are solution to the Cauchy problem (1.5), leads to the more realistic model
$$
\begin{equation}
\begin{split} &\frac{\partial v}{\partial t}+\sum_{i=1}^nv_i\, \frac{\partial v}{\partial x_i} -\mu_0\Delta v \\ &\qquad-\operatorname{Div}\int_0^tG(t-s)\mathcal{E}(v)(s, z(s,t,x))\,ds + \nabla p= f,\qquad (t,x)\in Q_T, \end{split}
\end{equation}
\tag{3.6}
$$
$$
\begin{equation}
z(\tau; t, x)=x+\int_t^\tau v(s, z(s; t, x))\,ds, \qquad 0\leqslant t, \tau\leqslant T, \quad x\in\overline{\Omega},
\end{equation}
\tag{3.7}
$$
$$
\begin{equation}
v(0, x)=v^0(x),\quad x\in \Omega, \qquad v(t, x)=0, \quad (t, x)\in [0, T]\times \partial\Omega.
\end{equation}
\tag{3.9}
$$
Our goal is to establish the solvability of problem (3.6)–(3.9) in the function class $v\in L_2(0,T;V)$. In this case, solvability of the Cauchy problem (3.7) is far from being obvious. This difficulty can be circumvented by employing the concept of a regular Lagrangian flow (RLF), which is a generalization of the concept of a classical solution. Let us recall some facts about RLFs. § 4. Regular Lagrangian FlowsLet $v\colon [0, T]\times \overline{\Omega}\to \mathbb{R}^N$. Consider the Cauchy problem (in the integral form)
$$
\begin{equation}
z(\tau; t, x)=x+\int_t^\tau v(s, z(s; t, x))\,ds, \qquad 0\leqslant t, \tau\leqslant T, \quad x\in\overline{\Omega}.
\end{equation}
\tag{4.1}
$$
For the definition of an RLF, see, for example, [38]–[40]. Here, we give this definition in the special case of a bounded domain $\Omega$ and for a divergence free $v$ that vanishes on the boundary. Definition 1. Let $v\in L_1(0, T;V)$. A regular Lagrangian flow (RLF) generated by $v$ is a function $z(\tau; t, x)$, $z\colon [0, T]\times [0, T]\times \overline{\Omega}\to \overline{\Omega}$, satisfying the following conditions: 1) for almost all $x$ and any $t\in[0, T]$, the function $\zeta(\tau)=z(\tau; t, x)$ is absolutely continuous and satisfies equation (4.1) and the condition $z(t; t, x)=x$; 2) for all $\tau, t \in[0, T]$, 3) for all $t_1, t_2,t_3\in[0, T]$ and almost all $x\in\overline{\Omega}$, Here, $B\subset\overline{\Omega}$ is an arbitrary Lebesgue measurable set, and $m(B)$ stands for the Lebesgue measure of $B$. Note that the existence of a unique RLF for the field $v$ is not guaranteed in general, the unique solvability of the Cauchy problem (3.7) in the classical sense (see [41]). The following results on RLF are worth recalling (see, for example, [40]). Theorem 1. Let $v\in L_1(0, T; W_{p}^1(\Omega)^N)$, let $1\leqslant p\leqslant +\infty$, let $\operatorname{div}v(t, x)=0$, and let $v(t, x)|_{\partial\Omega}=0$. Then there exists a unique RLF $z$ generated by $v$. Theorem 2. Let $v,v^n\in L_1(0, T; W_{1}^p(\Omega)^N)$, $n=1, 2, \dots$, for some $p>1$. Let $\operatorname{div}v^n=0$, $v^n|_{\partial\Omega}=0$, $v|_{\partial\Omega}=0$. Assume that Let $v^n$ converge to $v$ in $L_1(Q_T)^N$, and let $z^n(\tau; t, x)$ be the RLF generated by $v^n$. Then the sequence $z^n(\tau; t, x)$ converges to $z(\tau; t, x)$, which is the RLF generated by $v$, by $(\tau, x)$-measure uniformly with respect to $t\in [0, T]$. Here, $v_x$ is the Jacobi matrix of the vector function $v$. For a more general form of these results, see §§ 3.6, 3.7, 3.9 in [39]. Let us proceed with the main result. § 5. The main resultConsider the function space
$$
\begin{equation*}
W_1(0,T)\equiv \{v\colon v\in L_2(0,T;V) \cap L^{\infty}(0,T;H), \, v'\in L_1(0,T;V^{-1})\}.
\end{equation*}
\notag
$$
Let $v\in W_1(0,T)$. Theorem 1 implies the existence of a unique RLF $z(s, t,x)$ generated by $v$. Given a fixed $\tau,t\in [0,T]$, we introduce the operator $Z_v[\tau,t]$ associating with a function $v\in W_1(0,T)$ the function $Z_v[\tau,t](x)$ of the variable $x$ by $Z_v[\tau,t](x)=z(\tau, t,x)$. Definition 2. Let $ f\in L_1(0,T;V^{-1})$, $v^0\in H$. A weak solution to the problem (3.6)–(3.9) is said to be a function $v\in W_1(0,T)$ satisfying, for all $\varphi\in V$ and almost all $t\in[0,T]$ and the initial condition in (3.9), The inner product in the third and fourth terms in (5.1) is taken in the Hilbert space $L_2(\Omega)^{N\times N}$, and is defined by $ (A,B)= \sum_{i,j=1}^N\int_{\Omega}a_{ij}b_{ij}\,dx$ for any matrix functions $A,B\in L_2(\Omega)^{N\times N}$ with coefficients $a_{ij}$ and $b_{ij}$, respectively. Remark 1. Note the correctness of the integral term in (5.1). Indeed, from the estimate of the function $G$ in Proposition 1 and the obvious inclusion $\mathcal{E}(v)(s, y)\,ds\in L_2(0,T;L_2(\Omega)^{N\times N})$ we have $\int_0^tG(t-s)\mathcal{E}(v)(s, y)\,ds\in L_2(\Omega)^{N\times N}$. Note that the first factor $\int_0^t G(t-s)\,\mathcal{E}(v)(s, z(s;t,x))\,ds$ in the inner product in the integral term in (5.1) is obtained from the previous expression by the change of variable $y=z(s, t,x)$. From properties (4.2) and (4.3) of RLFs (see Lemma 3 below) we get the inclusion $\int_0^tG(t-s)\,\mathcal{E}(v)(s, z(s;t,x))\,ds\in L_2(\Omega)^{N\times N}$. Obviously, the second factor $\mathcal{E}(\varphi)$ lies in $L_2(\Omega)^{N\times N}$, and the inner product in the integral term makes sense. Remark 2. It is known that $W_1(0,T)\subset C_{\mathrm{weak}}([0,T],H)$ (see Theorem III.3.1 in [35]). Therefore, the initial condition from (3.9) is understood as the relation $\lim_{t\to 0}(v(t),\varphi)=(v^0,\varphi)$ for any $\varphi\in H$. The following result is the main one. § 6. Regularized problemsTo prove Theorem 3, we first construct a sequence of approximative regularized problems for problem (3.6)–(3.9). Let $\mathcal {P}$ be the orthogonal projection operator from the space $L_2(\Omega)^N$ onto $H$, and let $S\colon D(S) \to H $ be the Stokes operator, that is, $S(u)=- \mathcal {P} \Delta u$, $u\in D(S)= W_2^2(\Omega)^N\cap\mathring{W}_2^1(\Omega)^N\cap H$. The operator $S$ is a positive self-adjoint operator in $ H $ (see § II.4 in [42]). Consider the sequence of regularized problems depending on the numerical parameter $ n=1,2, \dots$:
$$
\begin{equation}
\begin{split} &\frac{\partial v^n}{\partial t}+\sum_{i=1}^N v_i^n\, \frac{\partial((1+ n^{-1}|v^n|^2)^{-1} v^n)}{\partial x_i} - \mu_0 \operatorname{Div}\mathcal{E} (v^n) \\ &\ -\operatorname{Div} \int_0^{t} G \biggl(t\,{-}\, \tau\,{+}\, \frac1{n}\biggr) \mathcal{E}(v^n)(\tau, z^n(\tau; t, x)) \, d\tau\,{+}\, \nabla p^n\,{=}\,f, \qquad (t, x)\,{\in}\, Q_T; \end{split}
\end{equation}
\tag{6.1}
$$
$$
\begin{equation}
v^n (0, x)=v^0 (x), \qquad x \in \Omega, \quad v^n |_{[0, T] \times \partial \Omega}=0;
\end{equation}
\tag{6.3}
$$
$$
\begin{equation}
z^n(\tau; t, x)=x+\int_t^\tau \widetilde v^{\,n}(s, z^n(s;t, x))\,ds, \qquad 0\leqslant t, \tau\leqslant T, \quad x\in\overline{\Omega}.
\end{equation}
\tag{6.4}
$$
Here, $|v|=\bigl(\sum_{i=1}^N v_i^2\bigr)^{1/2}$ is the norm of a vector $v$ in $\mathbb{R}^N$, the operator $ \widetilde{\ }$ (tilde) is defined by $\widetilde u=S_n u$ for $u\in V$, so that
$$
\begin{equation}
\widetilde v^{\,n}=S_n v^n,\qquad S_n=(I+n^{-1}S)^{-1},
\end{equation}
\tag{6.5}
$$
and $z^n$ is the RLF generated by the vector field $\widetilde v^{\,n}$. Note that the resolvent inequality $|(nI+S)^{-1}v|_0 \leqslant Mn^{-1} |v|_0$ implies the easily verifiable inequalities
$$
\begin{equation}
|S_nv|_0\leqslant M |v|_0,\quad |S_nv|_1\leqslant M |v|_1,\quad |S_nv|_{-1}\leqslant M |v|_{-1},\qquad v\in V,
\end{equation}
\tag{6.6}
$$
which are uniform with respect to $ n=1,2,\dots$ . Let us introduce the function space
$$
\begin{equation*}
W(0,T)\equiv \{v:\ v\in L_2(0,T;V)\cap L^{\infty}(0,T;H), \ v'\in L_2(0,T;V^{-1})\}.
\end{equation*}
\notag
$$
The weak solution $v^n\in W(0,T)$ to problem (6.1)–(6.4) is defined as a function $v^n \in W(0,T)$ satisfying the initial condition from (6.3) and the identity
$$
\begin{equation}
\begin{aligned} \, &\frac{d(v^n, \varphi)}{dt}-\sum_{i=1}^N\biggl(v_i^n( 1+n^{-1}|v^n|^2)^{-1}v^n, \frac{\partial \varphi}{\partial x_i}\biggr) +\mu_0(\mathcal{E}(v^n), \mathcal{E}(\varphi)) \nonumber \\ &\qquad+\biggl(\int_0^{t}G\biggl(t-\tau+\frac1{n}\biggr) \mathcal{E}(v^n)(\tau, Z_{\widetilde v^{\,n}}[\tau,t])\,d\tau, \mathcal{E}(\varphi)\biggr) =\langle f,\varphi\rangle \end{aligned}
\end{equation}
\tag{6.7}
$$
for any $\varphi \in V$ and almost all $ t\in [0, T]$. Here, the operator $Z_{\widetilde v^{\,n}}[\tau,t]$ associates with a function $\widetilde v^{\,n}$ the function $Z_{\widetilde v^{\,n}}[\tau,t](x)$ of $x$ by the rule $Z_{\widetilde v^{\,n}}[\tau,t](x)=z^n(s, t,x)$, where $z^n(s, t,x)$ is the RLF generated by the function $\widetilde v^{\,n}$. Since $v^n \in W (0, T) $, it follows from (6.6) that $\widetilde v^{\,n} \in W(0,T)$, and by Theorem 1 there exists an RLF generated by $\widetilde v^{\,n}$. Theorem 4. Let $f\in L_2(0,T;V^{-1})$, $v^0 \in H$. Then for any $n=1,2, \dots$ there exists a weak solution $v^n \in W(0,T)$ to problem (6.1)–(6.4), and where the constant $M_0$ is independent of $n$. 6.1. Proof of Theorem 4The weak solvability of the regularized problems (6.1)–(6.4) for fixed $n$ follows from [19]. Note that [19] was concerned with the case of an exponential type kernel $G(t-\tau)$ and another type of regularization $\widetilde v^{\,n}$. But this is immaterial, since the regularized kernel $G(t-\tau+1/n)$ in the present case is a smooth function, and the regularization $\widetilde v^{\,n}$ has the necessary properties (see, for example, [20]). Let us prove estimates (6.8), (6.9). Let us write problem (6.1)–(6.4) in an operator form. It will be convenient for us to treat $v$ as a function of $t$ with values in $H$ (and denote it by $v(t)$); let $v'(t)$ be its derivative. Using (6.7), we define the operators $B$ and $K_\varepsilon$ from $V$ to $V^{-1}$
$$
\begin{equation}
\begin{gathered} \, \langle B (u), h \rangle = \bigl(\mathcal{E} (u), \mathcal{E} (h)\bigr)_{L_2(\Omega)^{N \times N}}, \qquad u, h \in V; \\ \nonumber \langle K_\varepsilon(u), h \rangle=\sum_{i, j=1}^N \biggl(u_iu_j (1+ \varepsilon |u|^2)^{-1}, \, \frac{\partial h_i}{\partial x_j}\biggr)_{L_2 (\Omega)}, \qquad \varepsilon \geqslant 0, \quad u,h \in V. \end{gathered}
\end{equation}
\tag{6.10}
$$
We set
$$
\begin{equation}
G_{k,n}(s)=\begin{cases} \exp(-ks)G\biggl(s+\dfrac1{n}\biggr), &s\geqslant 0, \\ 0, &s<0. \end{cases}
\end{equation}
\tag{6.11}
$$
Note that the kernel $G_{k,n}(s)$ is continuously differentiable for $s\geqslant 0$. For $v,u\in L_2(0,T;V)$ and every fixed $t \in (0,T)$, we introduce the functional $C_{k,n}(u,Z_{v})\in V^{-1}$ acting on $h\in V$ by
$$
\begin{equation}
\langle C_{k,n}(u,Z_{ v}), h \rangle = \biggl( \int_0^{t} G_{k,n}(t-\tau,n) \mathcal{E}(u)(\tau, Z_{\widetilde v}[\tau,t]) \, d \tau, \mathcal{E} (h)\biggr)_{L_2(\Omega)^{N \times N}}, \qquad h \in V.
\end{equation}
\tag{6.12}
$$
Here, the operator $Z_{\widetilde v}[\tau,t]$ associates with a function $\widetilde v$ the function $Z_{\widetilde v}[\tau,t](x)$ of $x$ by the rule $Z_{\widetilde v}[\tau,t](x)=z(s, t,x)$, where $z(s, t,x)$ is the RLF generated by the function $\widetilde v$. Note that, for any $\varepsilon>0$ (see [19]),
$$
\begin{equation}
\langle K_{\varepsilon}(v),v\rangle=(K_{\varepsilon}(v),v)=0,\qquad v\in V.
\end{equation}
\tag{6.13}
$$
Each $v \in W_1(0, T)$ satisfies (see § III, Lemma 1.1 in [35])
$$
\begin{equation*}
\langle v '(t), \varphi \rangle=\frac{d} {dt} (v (t), \varphi ) \quad \forall\, \varphi \in V.
\end{equation*}
\notag
$$
Now problem (6.1)–(6.4) can be written as (see [19])
$$
\begin{equation}
(v^n)'+K_{1/n}(v^n)+\mu_0 B(v^n)+ C_{0,n}(v^n,Z_{v^n}(v^n))=f,\qquad t\in[0,T],\quad v(0)=v^0.
\end{equation}
\tag{6.14}
$$
Let us estimate solutions to problem (6.14). It will be convenient for us to consider a problem more general than (6.14). Multiplying formally equation (6.14) by $\exp(-kt)$, where $k>0$, we have, after some algebra, get the problem
$$
\begin{equation}
v'+\exp(-kt)K_{1/n}( v)+ k v+\mu_0 B(v)+C_{k,n}(v,Z_{v^n})= F,\qquad t\in[0,T], \quad v(0)=v^0.
\end{equation}
\tag{6.15}
$$
Here, we put, for brevity, $v=\exp(-kt)v^n$, $F=\exp(-kt)f$. Let us establish estimates of solutions to problem (6.15). Lemma 1. Let $k\geqslant k_0$, where $k_0$ is large enough. Then, for any weak solution $v$ to problem (6.15), where $M_0$ is independent of $n$. Proof. To prove estimate (6.16), we apply both parts of equation (6.15) (as a functional of $V^{-1}$) to $v$. Using (6.13), we have, after some simple algebra, we get
It follows that
$$
\begin{equation}
\frac{d}{dt}| v(t)|_0^2 +|v|_1^2\leqslant M\biggl(|v(t)|_1|F(t)|_{-1}+ |v(t)|_1 \int_0^tG_{k,n}(t-s)\,|v(s)|_1\,ds\biggr).
\end{equation}
\tag{6.17}
$$
From (6.17) we have for an arbitrary $\delta>0$
$$
\begin{equation}
\frac{1}{2}\, \frac{d}{dt} |v(t)|_0^2 +|v(t)|_1^2\leqslant \delta|v(t)|_1^2 +C_1(\delta)|F(t)|_{-1}^2+ C_1(\delta) \biggl(\int_0^tG_{k,n}(t-s)|v(s)|_1\,ds\biggr)^2.
\end{equation}
\tag{6.18}
$$
Choosing $\delta$ small enough and transferring the first term from the right-hand side to the left-hand side in (6.18), we get
$$
\begin{equation}
\frac{d}{dt} |v(t)|_0^2 +|v(t)|_1^2\leqslant M_1\biggl(|F(t)|_{-1}^2 + \biggl(\int_0^tG_{k,n}(t-s)|v(s)|_1\,ds\biggr)^2\biggr).
\end{equation}
\tag{6.19}
$$
Integrating (6.19) over $[0,t]$, $0\leqslant t\leqslant T$, we get
$$
\begin{equation}
|v(t)|_0^2 +\int_0^t|v(s)|_1^2\,ds \leqslant M_2\biggl(\|F\|_{0,-1}^2 +|v^0|_0^2 + \int_0^t\biggl(\int_0^{\xi}G_{k,n}(\xi-s)|v(s)|_1\,d\xi\biggr)^2\,ds\biggr).
\end{equation}
\tag{6.20}
$$
We set Extending $v$ with zero outside the interval $[0,T]$ and changing the variable $\xi=t-s$, we rewrite the right-hand side part of (6.21) as
$$
\begin{equation}
\begin{aligned} \, g(t) &=\int_{-\infty}^{+\infty} G_{k,n}(t-s)|v(s)|_1\,ds \nonumber \\ &=\int_{-\infty}^{+\infty}G_{k,n}(\xi)|v(t-\xi)|_1\,d\xi = \int_{-\infty}^{+\infty}G_{k,n}(\xi)|v(t-\xi )|_1\,d\xi. \end{aligned}
\end{equation}
\tag{6.22}
$$
Using the integral Minkowski inequality, the invariance of the $L_2(-\infty,\infty)$-norm with respect to shifts, we obtain the inequality
$$
\begin{equation}
\begin{aligned} \, &\|g\|_{L_2(-\infty,\infty)} \leqslant \int_{-\infty}^{+\infty}G_{k,n}(\xi)\||v(t-\xi )|_1 \|_{L_2(-\infty,\infty)}\,d\xi \nonumber \\ &\qquad\leqslant\int_{-\infty}^{+\infty}G_{k,n}(\xi)\,d\xi\, \||v(t)|_1 \|_{L_2(-\infty,\infty)} = \int_{-\infty}^{+\infty}G_{k,n}(\xi)\,d\xi\, \||v(t)|_1 \|_{L_2(0,T)}. \end{aligned}
\end{equation}
\tag{6.23}
$$
From (6.11) and (8.19) we have $G_{k,n}(s)=0$ for $s\leqslant 0$, and
$$
\begin{equation}
G_{k,n}(s)\leqslant M\exp(-ks)\biggl(s+\frac1{n}\biggr)^{\gamma_1-1} \quad \text{for}\quad s>0.
\end{equation}
\tag{6.24}
$$
It follows that
$$
\begin{equation}
\int_{-\infty}^{+\infty}G_{k,n}(\xi)\,d\xi=\int_0^{+\infty}G_{k,n}(\xi)\,d\xi \leqslant Mk_0^{-\gamma_1}.
\end{equation}
\tag{6.25}
$$
From (6.23), (6.25) and since $\||v(t)|_1 \|_{L_2(0, T)}\leqslant \|v\|_{0,1}$, we have
$$
\begin{equation}
\|g\|_{L_2(0, T)}\leqslant M_4k_0^{-\gamma_1}\|v\|_{0,1}.
\end{equation}
\tag{6.26}
$$
Using estimates (6.20) and (6.26), this gives
$$
\begin{equation}
\sup_{0\leqslant t\leqslant T}| v(t)|_0^2 +\|v\|_{0,1}^2 \leqslant M_4(\|F\|_{0,-1}^2 +|v^0|_0^2)+M_4k_0^{-\gamma_1}\|v\|_{0,1}.
\end{equation}
\tag{6.27}
$$
Choosing $k_0$ large enough so that $M_4k_0^{-\gamma_1}\leqslant q<1$ and transferring the last term in (6.27) to the left-hand part, we get estimate (6.16).
Lemma 1 is proved. Lemma 2. Let $k_0$ be large enough, $k\geqslant k_0$. Then, for any for weak solutions $v$ to problems (6.15), where $ M_0 $ is independent of $n$. Proof. Let us verify estimate (6.28). Let $v\in W_1(0,T)$ be a weak solution to problem (6.15). Then
$$
\begin{equation}
v '= F -\exp(-kt)K_{1/n}( v) -k v-\mu_0 B(v) - C_{k,n}(v,Z_{\widetilde v^{\,n}}).
\end{equation}
\tag{6.29}
$$
Therefore,
$$
\begin{equation}
\begin{aligned} \, &\|v '\|_{L_1 (0,T;V^{-1})} \leqslant \|F\|_{L_1 (0,T;V^{-1})} +\|\exp(-kt)K_{1/n}( v)\|_{L_1 (0,T;V^{-1})} \nonumber \\ &\qquad+\|k v\|_{L_1 (0,T;V^{-1})} + \mu_0 \|\operatorname{Div}\mathcal{E}(v)\|_{L_1 (0,T;V^{-1})} \nonumber \\ &\qquad+\biggl\|\operatorname{Div}\int_0^tG_{k,n}(t-s) \mathcal{E}(v)(s, Z_{\widetilde v^{\,n}}[s,t])\,ds\biggr\|_{L_1(0,T;V^{-1})}. \end{aligned}
\end{equation}
\tag{6.30}
$$
In [19], it was shown that, for any $n>0$,
To proceed the proof of Lemma 2, we need an estimate of the last term in (6.30). Proof. It is easy to show that if an ($N\times N$) matrix $\mathcal{A}(x)$ belongs to $L_2 (\Omega)^{N\times N}$, then (see, for example, [18])
$$
\begin{equation}
|{\operatorname{Div}\mathcal{A}(x)}|_{-1} \leqslant M | \mathcal{A}(x)|_0.
\end{equation}
\tag{6.34}
$$
In view of (6.34) we have
$$
\begin{equation}
\begin{aligned} \, &\biggl| \operatorname{Div}\int_0^tG_{k,n}(t-s) \mathcal{E}(v)(s, Z_{\widetilde v}[s,t])\,ds\biggr|_{-1} \leqslant M\biggl|\int_0^tG_{k,n}(t-s) \mathcal{E}(v)(s, Z_{\widetilde v}[s,t])\,ds\biggr|_0 \nonumber \\ &\qquad\leqslant M \int_0^tG_{k,n}(t-s) |\mathcal{E}(v)(s, Z_{\widetilde v}[s,t])|_0\,ds. \end{aligned}
\end{equation}
\tag{6.35}
$$
Let us estimate $|\mathcal{E}(v)(s, Z_{\widetilde v^{\,n}}[s,t])|_0^2 =\int_{\Omega} |\mathcal{E}(v)(s, z(s;t,x))|^2\, dx$, where $z(s, t,x)$ is the RLF generated by $\widetilde v$.
Proof. Let $\chi_{B}$ be the indicator function of a measurable set $B\subset\Omega$. Since $v$ is divergence free, it follows from the properties (4.2), (4.3) in Definition 1 that Hence $\int_{\Omega}g(y)\,dy=\int_{\Omega}g(z (s; t, x))\,dx$, and, for any summable function $g$ which corresponds to the change of the variable $y=z(s; t, x)$ in the first integral.
Therefore, changing the variable $x=z(t; s, y)$ in the first integral in (6.36) and using the equality $z(s;t,z(t; s, y))=y$, which follows from (4.3), we get
$$
\begin{equation*}
\int_{\Omega} |\mathcal{E}(v)(s, z(s;t,x))|^2 \, dx =\int_{\Omega} |\mathcal{E}(v)(s, z(s;t,z (t; s, y)))|^2 \, dx =\int_{\Omega} |\mathcal{E}(v)(s, y))|^2 \, dy
\end{equation*}
\notag
$$
or, what is the same, formula (6.36).
Lemma 3 is proved. It is plain that
$$
\begin{equation}
\int_{\Omega} |\mathcal{E}(v)(s, y))|^2 \, dy \leqslant M \sum_{i, j=1}^N\int_{\Omega} \biggl| \frac{\partial v_i(s, x)}{\partial x_j}\biggr|^2 \, dx \leqslant M|v(t)|_1.
\end{equation}
\tag{6.37}
$$
Hence, by (6.35), (6.37),
$$
\begin{equation}
\biggl| \operatorname{Div}\int_0^tG_{k,n}(t-s) \mathcal{E}(v)(s, Z_{\widetilde v}[s,t])\,ds \biggr|_{-1} \leqslant M \int_0^tG_{k,n}(t-s)|v(s)|_1\,ds.
\end{equation}
\tag{6.38}
$$
Inequality (6.33) follows from (6.21), (6.26) and (6.38). Proposition 2 is proved. From (6.31)–(6.33), (6.16) it follows that all the terms of the right of equation (6.29) belong to $L_1(0, T; V^{-1})$, and hence $v' \in L_1 (0,T;V^{-1})$. Let us set estimate (6.28). From estimates (6.29), (6.31)–(6.33), (6.16) and the fact that the $L_p{(0,T;V^{-1})}$-norm is majored by $L_q{(0,T;V^{-1})}$ for $q > p$, it follows that each term on the right side of equation (6.29) belongs to $L_2(0, T; V^{-1})$, and, in addition, the inequality holds
$$
\begin{equation*}
\begin{aligned} \, \|v'\|_{L_1(0, T; V^{-1})} &\leqslant \|F\|_{L_1(0, T; V^{-1})}+\|K_{1/n}(v)\|_{L_1(0, T; V^{-1})} + \|{\operatorname{Div}\mathcal{E}(v)}\|_{L_1 (0,T;V^{-1})} \\ &\qquad+\biggl\| \operatorname{Div} \int_0^tG_{k,n}(t-s)\mathcal{E}(v)(s, Z_{\widetilde v^{\,n}}[s,t])\,ds \biggr\|_{L_1 (0,T;V^{-1})} \\ &\leqslant M(1+\|F\|_{L_2(0,T;V^{-1})}+|v|_0)^2. \end{aligned}
\end{equation*}
\notag
$$
The proves estimate (6.28), and, therefore, Lemma 3. To complete the proof of Theorem 4, it remains to verify estimates (6.8) and (6.9). It is clear that $v^n=\exp(kt)v$, $z=z^n$, $f=\exp(kt)F$ and $(v^n)'=\exp(kt)v'+k\exp(kt)v$. Hence from inequalities (6.16) and (6.9) we get estimates (6.8) and (6.9) with the constant $M_0$ independent of $n$, which but depends on $k$ and $M$. Theorem 4 is proved. § 7. Proof of Theorem 3It follows from Theorem 4 that there is a subsequence $v^{n_k}$, $k=1,2,\dots$, of the sequences $v^n$, $n=1,2,\dots$, that weakly converges to some $v\in W_1(0,T)$ in $L_2(0,T;V)$, weakly$^*$ converges in $L_{\infty}(0,T;H)$, strongly converges in $L_2(0,T;H)$, and $(v^{n_k})'$ converges to $v'$ in the sense of distributions. Without loss of generality we assume that this subsequence coincides with the original sequence $v^n$. Lemma 4. The convergence of $v^n$ to $v$ in $L_2(0,T;H)$ implies that of $\widetilde v^{\,n}$ to $v$ in $L_2(0,T;H)$. Proof. It is clear that
$$
\begin{equation}
\widetilde v^{\,n}-v=S_n v^n-v=S_n( v^n-v)+ ( S_n-I) v.
\end{equation}
\tag{7.1}
$$
Let us write $v^n-v$ as $v^n-v=\mathcal{M}_1(n)-\mathcal{M}_2(n)$, where
$$
\begin{equation*}
\mathcal{M}_1(n)=S_n( v^n-v),\qquad \mathcal{M}_2(n)=( S_n-I) v.
\end{equation*}
\notag
$$
It follows from (6.6) that $\|\mathcal{M}_1(n)\|_0\leqslant M \|v^n-v\|_0$ and, therefore, Note that we can consider $S_n$ as a bounded operator in $L_2(0,T;H)$. Further, it is well known that the set of functions $X=L_2(0,T;{W_2 ^2}(\Omega)^N\cap H)$ is dense in $L_2(0,T;H)$.
Let $\varphi\in X$, so that $S\varphi\in L_2(0,T;H)$. Then, by (6.6) we have
$$
\begin{equation*}
|(S_n-I)\varphi|_0= n^{-1}|(I+n^{-1}S)^{-1}S\varphi|_0\leqslant M n^{-1}|S\varphi|_0.
\end{equation*}
\notag
$$
This gives
$$
\begin{equation}
\lim_{n\to +\infty}\|(S_n-I)\varphi\|_0=0,\qquad \varphi\in X.
\end{equation}
\tag{7.3}
$$
The norms of the operators $S_n-I$ in $L_2(0,T;H)$ are, clearly, uniformly bounded, it follows from (7.3) that,for $\varphi\in X$ gives (see § III.4 in [43]), relation (7.3) holds for any $\varphi\in L_2(0,T;H)$. In particular, for $\varphi =v$, we have
$$
\begin{equation}
\lim_{n\to +\infty}\|\mathcal{M}_2(n)\|_0=\lim_{n\to +\infty}\|(S_n-I)v\|_0=0.
\end{equation}
\tag{7.4}
$$
Now the conclusion of Lemma 4 follows from (7.1), (7.2) and (7.4). Integrating (6.7) over $[0,T]$, we find that
$$
\begin{equation}
\begin{aligned} \, &(v^n(T), \varphi)-\sum_{i=1}^N\int_0^T\biggl( v_i^n (1+n^{-1}|v^n|^2)^{-1}v^n, \frac{\partial\varphi}{\partial x_i}\biggr)\,ds +\mu_0\int_0^T(\mathcal{E}(v^n), \mathcal{E}(\varphi))\,ds \nonumber \\ &\qquad\qquad + \int_0^T\biggl(\int_0^{t}G\biggl(t-s+\frac1{n}\biggr) \mathcal{E}(v^n)(s, Z_{\widetilde v^{\,n}}[s,t])\,ds, \mathcal{E}(\varphi)\biggr)\,dt \nonumber \\ &\qquad=\int_0^T \langle f, \varphi\rangle\,ds+(v^0, \varphi) \end{aligned}
\end{equation}
\tag{7.5}
$$
for any $\varphi \in V$ and almost all $t\in [0,T]$. Let us pass to the limit in (7.5). We set
$$
\begin{equation*}
\begin{aligned} \, \mathcal{P}_1(n) &=(v^n(T), \varphi), \\ \mathcal{P}_2(n) &=\sum_{i=1}^N\int_0^T\biggl( v_i^n (1+n^{-1}|v^n|^2)^{-1}v^n, \frac{\partial\varphi}{\partial x_i}\biggr)\,ds, \\ \mathcal{P}_3(n) &=\int_0^T(\mathcal{E}(v^n), \mathcal{E}(\varphi))\,ds, \\ \mathcal{P}_4(n) &=\int_0^T\biggl(\int_0^{t}G\biggl(t-s+\frac1{n}\biggr) \mathcal{E}(v^n)(s, Z_{\widetilde v^{\,n}}[s,t])\,ds, \mathcal{E}(\varphi)\biggr)\,dt. \end{aligned}
\end{equation*}
\notag
$$
Next, we rewrite (7.5) as
$$
\begin{equation}
\mathcal{P}_1(n)-\mathcal{P}_2(n)+\mu_0\mathcal{P}_3(n)+ \mathcal{P}_4(n) =\int_0^T \langle f, \varphi\rangle\,ds+(v^0, \varphi).
\end{equation}
\tag{7.6}
$$
Assume that $\varphi$ is smooth. It follows from estimate (6.9) and Lemma 1.4 in [35] that the $H$-valued function $v(t)$ is weakly continuous, and $v^n(T)$ is bounded in $H$. Without loss of generality we can assume that $v^n(T)$ converges weakly to $v(T)$ in $H$ and $v^n$ converges weakly to $v$ in $L_2(0,T;V)$. Therefore,
$$
\begin{equation}
\begin{aligned} \, \mathcal{P}_1 &=\lim_{n\to\infty}\mathcal{P}_1(n)=(v(T), \varphi), \\ \mathcal{P}_3 &= \lim_{n\to\infty}\mathcal{P}_3(n) =\int_0^T(\mathcal{E}(v), \mathcal{E}(\varphi))\,ds. \end{aligned}
\end{equation}
\tag{7.7}
$$
Let us show that
$$
\begin{equation}
\mathcal{P}_2=\lim_{n\to\infty}\mathcal{P}_2(n) =\sum_{i=1}^N\int_0^T\biggl(v_iv, \frac{\partial\varphi}{\partial x_i}\biggr)\,ds.
\end{equation}
\tag{7.8}
$$
It is clear that
$$
\begin{equation*}
\begin{aligned} \, \mathcal{P}_2(n)-\mathcal{P}_2 &=\sum_{i=1}^N\int_0^T\biggl(v^n_iv^n(1+n^{-1}|v^n|^2)^{-1}-v_iv, \frac{\partial\varphi}{\partial x_i}\biggr)\,ds \\ &= \sum_{i=1}^N\int_0^T\biggl((v^n_i-v_i) v^n(1+n^{-1}|v^n|^2)^{-1}, \frac{\partial\varphi}{\partial x_i}\biggr)\,ds \\ &= \sum_{i=1}^N\int_0^T\biggl(v_i(v^n-v) (1+n^{-1}|v^n|^2)^{-1}, \frac{\partial\varphi}{\partial x_i} \biggr)\,ds \\ &= \sum_{i=1}^N\int_0^T\biggl(v_iv n^{-1}|v^n|^2 (1+n^{-1}|v^n|^2)^{-1}, \frac{\partial\varphi}{\partial x_i}\biggr)\,ds. \end{aligned}
\end{equation*}
\notag
$$
Since $\varphi$ is smooth, we have
$$
\begin{equation*}
\begin{aligned} \, |\mathcal{P}_{2}(n)-\mathcal{P}_{2}| &\leqslant M\biggl(\sum_{i=1}^N\int_0^T\int_\Omega |v^n_i-v_i|\,|v^n|\,dx\,ds +\sum_{i=1}^N\int_0^T\int_\Omega |v^n_i|\,|v^n-v|\,dx\,ds \\ &\qquad\qquad +\sum_{i=1}^N\int_0^T \int_\Omega n^{-1}|v_i|\,|v |\,|v^n|^2 (1+n^{-1}|v^n|^2)^{-1}\,dx\,ds\biggr) \\ &=M\bigl(\mathcal{P}_{21}(n)+\mathcal{P}_{22}(n)+\mathcal{P}_{23}(n)\bigr). \end{aligned}
\end{equation*}
\notag
$$
Next, it is plain that
$$
\begin{equation*}
\mathcal{P}_{21}(n)+\mathcal{P}_{22}(n) \leqslant M\|v^n-v\|_{ L_2 (0, T;H)}\|v^n\|_{ L_2 (0, T;H)}.
\end{equation*}
\notag
$$
From the convergence of $v^n$ to $v$ in $L_2(0, T;H)$ it follows that
$$
\begin{equation}
\mathcal{P}_{21}(n)+\mathcal{P}_{22}(n)\to 0 \quad\text{as} \quad n\to+\infty.
\end{equation}
\tag{7.9}
$$
Consider $\mathcal{P}_{23}(n)$. It is easy to see that the convergence of $v^n$ to $v$ in $L_2 (0, T;H)$ entails the convergence of $v^n(t,x)$ to $v(t,x)$ almost everywhere on $Q_T$. It follows that the integrand
$$
\begin{equation*}
\Phi^n(t,x)=n^{-1}|v_i(t,x)|\,|v (t,x)|\,|v^n(t,x)|^2 (1+n^{-1}|v^n(t,x)|^2)^{-1}
\end{equation*}
\notag
$$
in $\mathcal{P}_{23}(n)$ tends to zero as $n\to+\infty$ almost everywhere on $Q_T$. It is easily checked that $\Phi^n(t,x) \leqslant$$ M|v_i(t,x)|\,|v (t,x)|\equiv \Phi(t,x)$. Since $v\in L_2 (0, T;H)$, it follows from the Hölder inequality that $\Phi(t,x)\in L_1(Q_T)$. An application of the Lebesgue theorem gives
$$
\begin{equation}
\mathcal{P}_{23}(n)\to 0 \quad\text{as} \quad n\to+\infty.
\end{equation}
\tag{7.10}
$$
Relation (7.8) follows from (7.9) and (7.10). For arbitrary ($N\times N$) matrices $A$ and $B$ with coefficients $a_{ij}$ and $b_{ij}$, respectively, we set $A:B=\sum_{i,j=1}^N a_{ij}b_{ij}$. Consider $\mathcal{P}_4(n)$ and write it as
$$
\begin{equation*}
\mathcal{P}_4(n)=\int_0^T\int_0^tG\biggl(t-s+\frac1{n}\biggr)\, \int_\Omega\mathcal{E}(v^n)(s, z^n(s;t,x)):\mathcal{E}(\varphi)( x)\,dx\,ds\,dt.
\end{equation*}
\notag
$$
Here, $z^n(s,t,x)=Z_{\widetilde v^{\,n}}[s,t](x)$. Let us change the variables $y=z^n(s,t,x)$ (inverse change is $x=z^n(t, s, y)$). Using the fact that the field $v^n$ is divergence free and taking into account the properties (4.2), (4.3) of RLFs in Definition 1, we find that
$$
\begin{equation*}
\begin{aligned} \, \mathcal{P}_4(n) &=\int_0^T\int_0^tG\biggl(t-s+\frac1{n}\biggr) \int_\Omega\mathcal{E}(v^n)(s, z^n(s;t,x)):\mathcal{E}(\varphi)(x)\,dx\,ds\,dt \\ &=\int_0^T\int_\Omega\int_0^tG\biggl(t-s+\frac1{n}\biggr) \mathcal{E}(v^n)(s, y):\mathcal{E}(\varphi)(z^n(t;s,y))\,dy\,ds\,dt. \end{aligned}
\end{equation*}
\notag
$$
Using this relation and changing the order of integration, we have
$$
\begin{equation*}
\begin{aligned} \, \mathcal{P}_4(n) &= \int_0^T\int_\Omega\mathcal{E}(v^n)(s, y):\int_s^TG \biggl(t-s+\frac1{n}\biggr)\mathcal{E}(\varphi)(z^n(t, s, y))\,dt\,dy\,ds \\ &=\int_0^T\int_\Omega\mathcal{E}(v^n)(s, y):\psi^n(s, y)\,dy\,ds, \end{aligned}
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\psi^n(s, y)=\int_s^TG\biggl(t-s+\frac1{n}\biggr) \mathcal{E}(\varphi)(z^n(t, s, y))\,dt.
\end{equation*}
\notag
$$
Consider $\psi^n$. In view of the uniform estimates (6.9) and (6.6), we see that the sequence $\|\widetilde v^{\,n}\|_{L_2(0,T; W_2^1 (\Omega)^N)}$ is uniformly bounded. From Theorem 2 and Lemma 4 it follows that the sequence $z^n$ converges (up to a subsequence) with respect to the $(0, T]\times\Omega$-Lebesgue measure uniformly with respect to $t \in [0, T] $ to the RLF $z(\tau; t, x)$ generated by $v$. Since $\varphi$ is smooth, the sequence of bounded functions $\mathcal{E}(\varphi)(z^n(t, s, y))$ converges almost everywhere on $Q_T$ to a bounded function $\mathcal{E}(\varphi)(z(t, s, y))$. By virtue of Lebesgue’s Theorem, the uniformly bounded sequence $\psi^n(s, y)$ converges almost everywhere on $Q_T$ to the bounded function
$$
\begin{equation*}
\psi(s, y)=\int_s^TG(t-s)\mathcal{E}(\varphi)(z(t,s, y))\,\,dt.
\end{equation*}
\notag
$$
Thus, in the integrand
$$
\begin{equation}
\mathcal{P}_4(n)=\int_0^T\int_\Omega\mathcal{E}(v^n)(s, y):\psi^n(s, y)\,dy\,ds
\end{equation}
\tag{7.11}
$$
the first multiplier converges weakly in $L_2(Q_T)^{N\times N}$, and the second one converges almost everywhere in $Q_T$. This means that in (7.11) one can make $n\to +\infty$, which gives
$$
\begin{equation}
\begin{aligned} \, \mathcal{P}_4 &= \lim_{n\to+\infty}\mathcal{P}_4(n) =\int_0^T\bigl(\mathcal{E}(v)(s,y), \psi(s, y)\bigr)\,ds \nonumber \\ &= \int_0^T\int_s^TG(t-s)\bigl(\mathcal{E}(v)(s,y), \mathcal{E}(\varphi)(Z_{v}[t,s])\bigr)\,dt\,ds. \end{aligned}
\end{equation}
\tag{7.12}
$$
Changing the order of integration and making the change of variables $y=z(s,t,x)$, we get
$$
\begin{equation}
\mathcal{P}_4=\int_0^T\int_0^t\bigl(\mathcal{E}(v)(s, Z_{v}[s,t]), \mathcal{E}(\varphi)\bigr)\,ds\,dt.
\end{equation}
\tag{7.13}
$$
From the convergence of the terms $\mathcal{P}_i(n)$ from (7.6) (see (7.7), (7.8) and (7.12)) it follows that the function $v(t, x)$ satisfies the identity
$$
\begin{equation}
\begin{aligned} \, &(v(T), \varphi)-\sum_{i=0}^N \int_0^T\biggl(v_iv, \frac{\partial\varphi}{\partial x_i}\biggr)\,ds +\mu_0\int_0^T\bigl(\mathcal{E}(v)(t), \mathcal{E}(\varphi)\bigr)\,dt \nonumber \\ &\qquad+ \int_0^T\int_0^tG(t-s)\bigl(\mathcal{E}(v)(s, Z_{ v}[s,t])\,ds, \mathcal{E}(\varphi)\bigr)\,dt = \int_0^T \langle f, \varphi\rangle\,dt \end{aligned}
\end{equation}
\tag{7.14}
$$
for any smooth $\varphi$. Let $\varphi \in V$ be arbitrary. We choose a sequence of smooth $\varphi_m \in V$, $m=1,2,\dots$, such that $\varphi_m$ converges in $V$ to $\varphi$ as $m\to +\infty$. Assuming $\varphi=\varphi_m $ in (7.14) and making $m\to +\infty$, we get (7.14) for an arbitrary $\varphi \in V$. The passage to the limit is possible, since the convergence of $\varphi_m$ to $\varphi$ in $V$ follows from that of $\mathcal{E}(\varphi_m)$ to $\mathcal{E}(\varphi)$ in $L_2(\Omega)^{N\times N}$ and, in addition, the inner products in (7.14) are continuous with respect to their multipliers. It is obvious that identity (7.14) implies the identity
$$
\begin{equation}
\begin{aligned} \, &(v(t), \varphi)- \sum_{i=0}^N\int_0^t\biggl(v_iv, \frac{\partial\varphi}{\partial x_i}\biggr)\,ds + \mu_0\int_0^t\bigl(\mathcal{E}(v)(s), \mathcal{E}(\varphi)\bigr)\,ds \nonumber \\ &\qquad+ \int_0^t\int_0^\tau G(\tau -s)\bigl(\mathcal{E}(v)(s, Z_v[s,t])\,ds, \mathcal{E}(\varphi)\bigr)\,d\tau = \int_0^t \langle f, \varphi\rangle\,d\tau \end{aligned}
\end{equation}
\tag{7.15}
$$
for all $0\leqslant t \leqslant T$. Differentiating (7.15) with respect to $t$, we verify that $v$ satisfies identity (5.1). Let us show that $v\in W_1(0,T)$. We write (5.1) as
$$
\begin{equation}
\frac{d(v,\varphi)}{dt} - \sum_{i=1}^N \biggl(v_iv, \frac{\partial \varphi}{\partial x_i}\biggr) +\mu_0\bigl( (\mathcal{E}),\mathcal{E}(\varphi)\bigr) =\langle w, \varphi \rangle,
\end{equation}
\tag{7.16}
$$
where
$$
\begin{equation}
w=f -\mathcal{P}\operatorname{Div}\int_0^tG(t-s)\mathcal{E}(v)(s, Z_v[s,t])\,ds.
\end{equation}
\tag{7.17}
$$
It follows from estimates (6.33) that the second term in (7.17) belongs to the space $L_2(0,T;V^{-1})$, and, consequently, $w\in L_2(0,T;V^{-1})$. Thus, $v$ satisfies the equation The inequality $\| K_0 (v)\|_{L_1 (0, T; V^{-1})}\leqslant M\| v \|_{L_2(0,T;V)}^2$ follows from [19]. Using this fact, (7.18) and estimates (6.8), (6.9), we obtain the inequality
$$
\begin{equation}
\begin{aligned} \, \| v'\|_{L_1 (0, T; V^{-1})} &\leqslant M (1+ \|w\|_{ L_2 (0, T; V^{-1})}+|v^0|_0)^2 \nonumber \\ &\leqslant M (1+ \| f \|_{ L_2 (0, T; V^{-1})}+|v^0|_0)^2. \end{aligned}
\end{equation}
\tag{7.19}
$$
From (7.19) and inequalities (6.8), (6.9), we have $v\in W_1(0,T)$. Therefore, by (5.1), $v$ is a weak solution to problem (3.6)–(3.9). Theorem 3 is proved. § 8. Proof of Proposition 1From (3.3) we have
$$
\begin{equation}
\begin{gathered} \, I_t^{a_m}D_t^{a_m}\sigma=\sigma,\qquad I_t^{a_m}D_t^{a_m}\mathcal{E}(v) =\mathcal{E}(v),\qquad I_t^{a_m}D_t^{a_k}\sigma=I_t^{a_m-a_k}\sigma, \\ I_t^{a_m}D_t^{b_k}\mathcal{E}(v)=I_t^{a_m-b_k}\mathcal{E}(v). \end{gathered}
\end{equation}
\tag{8.1}
$$
Applying the operator $I_t^{a_m}$ to (3.1) and using (3.3)–(8.1), we get
$$
\begin{equation}
\sigma={\mu_0}\mathcal{E}(v)+ p_m^{-1} \sum_{k=0}^{m-1}q_kI_t^{a_m-b_k} \mathcal{E}(v) - p_m^{-1}\sum_{k=0}^{m-1}p_kI_t^{a_m-a_k}\sigma.
\end{equation}
\tag{8.2}
$$
Here, $\mu_0=p_m^{-1}q_m$, $p_0=1$, $a_0=0$, $q_0=\nu$, $b_0=0$. We set
$$
\begin{equation}
F= {\mu_0}\mathcal{E}(v)+ p_m^{-1} \sum_{k=0}^{m-1}q_kI_t^{a_m-b_k} \mathcal{E}(v),\qquad K(\sigma)=-p_m^{-1}\sum_{k=1}^{m-1}p_kI_t^{a_m-a_k}\sigma.
\end{equation}
\tag{8.3}
$$
Now (8.2) assumes the form We first consider equation (8.4) for $\sigma$ in $L_2(0,T;L_2(\Omega)^{N\times N})$. Let us show that this equation is solvable.
$$
\begin{equation}
\begin{aligned} \, K(\sigma) &= \int_0^t \biggl(-p_m^{-1}\sum_{k=0}^{m-1} p_k\Gamma^{-1}(a_m-a_k)(t-s)^{a_m-a_k-1}\biggr)\sigma(s)\,ds \nonumber \\ &=\int_0^t\biggl(-p_m^{-1}\sum_{k=0}^{m-1}p_k\Gamma^{-1}(t-s)^{a_{m-1}-a_k}\biggr) (t-s)^{a_m-a_{m-1}-1}\sigma(s)\,ds. \end{aligned}
\end{equation}
\tag{8.5}
$$
Setting
$$
\begin{equation}
G_0(t-s)=- p_m^{-1}\sum_{k=0}^{m-1}p_k\Gamma^{-1}(t-s)^{a_{m-1}-a_k},
\end{equation}
\tag{8.6}
$$
we see that the operator $K$ has the form
$$
\begin{equation}
K(\sigma)=\int_0^tG_0(t-s)(t-s)^{\gamma_0-1}\sigma(s)\,ds,\qquad \gamma_0=a_m-a_{m-1}.
\end{equation}
\tag{8.7}
$$
Here, $G_0(z)$ is a function continuous on $[0,T]$, and $\gamma_0\in (0,2)$. Thus, $K$ is a Volterra type operator whose kernel has a weak singularity at $\gamma_0\in (0,1)$. Let us show that the operator $I-K$ is invertible in $L_2(0,T;L_2(\Omega)^{N\times N})$. Assume that $\overline{y}=y\exp(-kt)$, where $k>0$, and $y$ is an arbitrary function from $L_2(0,T;L_2(\Omega)^{N\times N})$. Consider the operator $\overline K$ defined by
$$
\begin{equation}
\overline{K}\overline{\sigma} =\int_0^t\exp(k(s-t))G_0(t-s)(t-s)^{\gamma_0-1} \overline{\sigma}(s)\,ds.
\end{equation}
\tag{8.8}
$$
We have
$$
\begin{equation}
\begin{aligned} \, |\overline{K}\overline{\sigma}| &\leqslant M_0\int_0^t\exp(k(s-t))(t-s)^{\gamma_0-1}|\overline{\sigma}(s)|\,ds \nonumber \\ &=M_0\int_0^t\exp(-k\tau)\tau^{\gamma_0-1}|\overline{\sigma}(t-\tau)|\,d\tau. \end{aligned}
\end{equation}
\tag{8.9}
$$
Using (8.9) and the integral Minkowski inequality, it is not difficult to show that
$$
\begin{equation}
\begin{aligned} \, \|\overline{K}\overline{\sigma}\|_{L_2(0,T;L_2(\Omega)^{N\times N})} &\leqslant M_0\int_0^{+\infty}\exp(-k\tau)\tau^{\gamma_0-1}\,d\tau\, \|\overline{\sigma}\|_{L_2(0,T;L_2(\Omega)^{N\times N})} \nonumber \\ &=M_0k^{-\gamma_0}\int_0^{+\infty}\exp(-s)s^{\gamma_0-1}\,ds\, \|\overline{\sigma}\|_{L_2(0,T;L_2(\Omega)^{N\times N})}. \end{aligned}
\end{equation}
\tag{8.10}
$$
Recall that $\|\,{\cdot}\,\|_0$ denotes the norm in $L_2(0,T;L_2(\Omega)^{N\times N})$. Using (8.8) and choosing $k$ large enough, we get
$$
\begin{equation}
\|\overline{K}\overline{\sigma}\|_{L_2(0,T;L_2(\Omega)^{N\times N})} \leqslant \overline q\|\overline{\sigma}\|_{L_2(0,T;L_2(\Omega)^{N\times N})},
\end{equation}
\tag{8.11}
$$
where $0<\overline q<1$. Consider the equivalent norm $]y[=\|\overline{y}\|_{L_2(0,T;L_2(\Omega)^{N\times N})}$, $\overline{y}=y\exp(-kt)$, in $L_2(0,T;L_2(\Omega)^{N\times N})$. Let us rewrite the inequality (8.11) as It follows from (8.12) that operator $(I-K)$ has a bounded inverse $(I- K)^{-1}=\sum_{l=0}^{+\infty}K^l$, where convergence is understood in the sense of the operator norm induced by norm $]\,{\cdot}\,[$.Since the norms $]\,{\cdot}\,[$ and $\|\,{\cdot}\,\|_{L_2(0,T;L_2(\Omega)^{N\times N})}$ are equivalent, $(I- K)$ is invertible and the operator $(I-K)^{-1}$ is bounded in $L_2(0,T;L_2(\Omega)^{N\times N})$. It follows that equation (8.4) is uniquely solvable, and its solution has the form $\sigma=(I-K)^{-1}F$. Using the explicit form of $F$ from (8.3), we can write this solution as
$$
\begin{equation}
\sigma=F+\sum_{l=1}^{+\infty}K^l(F)=\mu_0\mathcal{E}(v) + \mathcal{Z}(\mathcal{E}(v)),
\end{equation}
\tag{8.13}
$$
where $\mathcal{Z}(\mathcal{E}(v)) =Z_1(\mathcal{E}(v))+Z_2(\mathcal{E}(v))+Z_3(\mathcal{E}(v))$ and
$$
\begin{equation}
\begin{aligned} \, Z_1(\mathcal{E}(v)) &=p_m^{-1}q_{m-1}I_t^{a_m-b_{m-1}} \mathcal{E}(v), \qquad Z_2(\mathcal{E}(v)) =\mu_0K^1(\mathcal{E}(v)), \\ Z_3(\mathcal{E}(v)) &=p_m^{-1}\sum_{k=0}^{m-2}q_kI_t^{a_m-b_k} \mathcal{E}(v) +\sum_{l=2}^{+\infty}\mu_0K^l(\mathcal{E}(v)) \\ &\qquad+ p_m^{-1}\sum_{l=1}^{+\infty} \sum_{k=0}^{m-1}q_k K^l (I_t^{a_m-b_k} (\mathcal{E}(v)). \end{aligned}
\end{equation}
\tag{8.14}
$$
It follows from (3.2) and (8.7) that the kernels $\overline G_1(t-s)$ and $\overline G_2(t-s)$ of the integral operators generated by the terms $Z_1$ and $Z_2$ satisfy the inequalities
$$
\begin{equation}
\begin{gathered} \, |\overline G_1(t-s)|\leqslant M(t-s)^{\gamma},\qquad |\overline G_2(t-s)|\leqslant M(t-s)^{\gamma_0}, \\ \gamma =a_m-b_{m-1},\qquad \gamma_0= a_m-a_{m-1}. \end{gathered}
\end{equation}
\tag{8.15}
$$
Let us move on to $Z_3$. Consider the operators $R_i(y)=\int_0^tQ_i(t-s)y(s)\,ds$, $i=1,2$, where $|Q_i(t-s)|\leqslant M_i(t-s)^{-\alpha_i}$, $\alpha_i<1$. It is not difficult to show that the superposition $R_3=R_1R_2$ of the operators $R_1$, $R_2$ has the form $R_3(y)=\int_0^tQ_3(t-s)y(s)\,ds$, $|Q_3(t-s)|\leqslant M_3(t-s)^{1-\alpha_1-\alpha_2}$. From this and (8.7) it follows that the kernels $G_r(t-s)$ of the operators $K^r$, $r=2,3,\dots$, satisfy the inequality $|G_r(t-s)|\leqslant M_r(t-s)^{r\gamma_0-1}$. Hence, for all $r$ exceeding some $r_0>0$, the kernels of operators $K^r$ are bounded, and the largest order of singularity of the operators $K^r$ is attained at $r=1$. Note that, for $\gamma_0\in [1,2)$, $\gamma_0=a_m-a_{m-1}$, the kernel of operator $K$ is bounded. From the same formulas (3.2), (8.7) and the above arguments, it follows that the operator $\mathcal{Z}$ is a Volterra type integral operator, which has the form
$$
\begin{equation}
\mathcal{Z}(\mathcal{E}(v))=\int_0^tG_1(t-s) \mathcal{E}(v) (s)\,ds,
\end{equation}
\tag{8.16}
$$
where $G_1(t)$ satisfies the estimate
$$
\begin{equation}
|G_1(t-s)|\leqslant M(t-s)^{\gamma_1-1},\qquad 0\leqslant s \leqslant T,\quad \gamma_1=\min (\gamma, \gamma_0).
\end{equation}
\tag{8.17}
$$
From (8.14)–(8.17) it follows that
$$
\begin{equation}
\sigma=\mu_0\mathcal{E}(v)+ \int_0^tG(t-s)\mathcal{E}(v) (s)\,ds,
\end{equation}
\tag{8.18}
$$
where the kernel $G(t-s)$ satisfies the inequality
$$
\begin{equation}
|G(t-s)|\leqslant M (t-s)^{\gamma_1-1},\qquad 0\leqslant s \leqslant t,\quad \gamma_1=a_m-b_{m-1}.
\end{equation}
\tag{8.19}
$$
Proposition 1 is proved. 
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