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On the classical solution of the macroscopic model of in-situ leaching of rare metals A. M. Meirmanov Moscow State University of Civil Engineering
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2022, Volume 86, Issue 4, DOI: https://doi.org/10.4213/im9144 
Bibliographic databases:
    § 1. IntroductionIn this paper we use the homogenization method for media with a special periodic structure in order to rigorously derive a macroscopic mathematical model of in-situ leaching of rare metals and prove the existence and uniqueness theorems for classical solutions of the initial-boundary value problem for the system of differential equations of the macroscopic mathematical model. Extraction of rare metals by leaching is a very important task of the national economy. Natural deposits of uranium, nickel and other rare metals are complex and geologically heterogeneous objects. Their inhomogeneity means that the properties of an object of interest vary in space. Analyses of wells and cores show that the geological properties (porosity, permeability and so on) of ore bodies are heterogeneous even within a single deposit. An insufficient account of the consequences of inhomogeneity at the stage of operation planning very often becomes obvious too late, when the acid solution which was uploaded to soil through injection wells appears to be far from the intended location. Moreover, an important role is played by the concentration of the injected acid, the injection modes of acid solutions, and other factors. Hence, understanding the dynamics of fluids in heterogeneous porous media and the mechanism of dissolution of rocks by acids is of a fundamental importance for efficient management of rare metals mining. It is achieved by creating a prototype of a hydrodynamical simulator for the ore body based on an appropriate mathematical model. This enables one to optimize the entire technological process. A hydrodynamical simulator of an ore body is a complex consisting of the following data: a scale of mathematical models ordered in accordance with the accuracy of approximation of the given physical process (the prototype of the hydrodynamical simulator), a set of numerical characteristics of the geometrical and physical properties of the solid skeleton in the given deposit, and a set of computer programs that enable us to visualize the physical process and determine the dynamical changes of the main characteristics of the mathematical model. Currently there is a large range of mathematical models describing the dynamics of rock leaching at the macroscopic level (see [1]–[3] and references therein). In contrast to microscopic models, the characteristic size of macroscopic models is several meters or tens of meters. Therefore these models do not distinguish the micro-structure of a continuous medium: they postulate a simultaneous presence, at every point of the medium, of the solid skeleton and a liquid in the pores or cracks of this skeleton. All these models are built based on the same principles. Fluid dynamics is usually controlled by Darcy’s system of filtration or a modification of it. The equations describing the migration of the acid and the chemical reaction products are simply postulated. Roughly speaking, they are modifications of the diffusion-convection equations for the corresponding concentrations. The main thing in these postulates is the choice of coefficients of the equations. Here we observe a great variety of models depending on the tastes and preferences of their authors. It is quite understandable since the main mechanism of the physical process is focused on the unknown (free) boundary between the pore space and the solid skeleton and is not spelled out in any way in the proposed macroscopic models. This is where the rocks dissolve, changing the concentration of the injected acid, and this is where the products appear inside the carrying liquid. Moreover, during the process, the geometry of the pore space (or of the boundary separating the solid skeleton and the pore space) varies in space and time. These fundamentally important changes occur at the microscopic level, which corresponds to the average size of pores or cracks in rocks, while all the proposed macroscopic models operate at a completely different scale (tens times larger) and, therefore, do not distinguish neither the free boundary nor any features of interaction between the acid and the rocks. This explains the large variety of macroscopic mathematical models. Their authors simply have no exact methods for describing the physical processes at the microscopic level on the base of the fundamental laws of continuum mechanics and chemistry. They also lack the possibility to take this micro-structure into account in the macroscopic models. Therefore they have to restrict themselves to certain speculative considerations. Given several macroscopic models describing the same physical process under the same conditions, we may ask which of them reflects this process most adequately. Where is the criterion of adequacy here? Appeal to experiment makes no sense since every such model has many free parameters unrelated to the geometry of the reservoir (for example, porosity) or to the physical characteristics of the process (for example, the viscosity and density of filtered liquids) and, varying these parameters, one can match with any experiment. Burridge and Keller [4] and Sanchez–Palencia [5] were the first to explain that an adequate description of physical processes at the macroscopic level in the case of acoustics and filtration of liquids in rocks with a periodic structure is possible if and only if (a) the physical process under consideration is described at the microscopic level by the classical Newtonian equations of continuum mechanics (exact model), (b) we select a set of small dimensionless parameters, (c) the macroscopic mathematical model is a rigorous asymptotic limit (homogenization) of the exact mathematical model at the microscopic level as the selected small parameters tend to zero. Various special cases of exact macroscopic models of acoustics and fluid filtration in rocks with a periodic structure were investigated by many authors; see [6]–[13] and others. All these authors used various methods of homogenization and solution of every problem (as a rule, difficult) required considerable effort and ingenuity. We recall that a periodic structure with periodicity cell of size $\varepsilon$ is the structure obtained by repeating a cube of size $\varepsilon$ consisting of a liquid component and a solid component. This structure is determined by an $\varepsilon$-periodic function $\chi^{\varepsilon}(\boldsymbol{x})$ which is equal to one (resp. zero) in the pore space filled with the liquid (resp. in the solid skeleton). The function $\chi^{\varepsilon}(\boldsymbol{x})$ is called the characteristic function of the structure. Everything changed after the appearance of Nguetseng’s paper [14], which suggested the method of two-scale convergence in periodic structures. Things that used to be an art has become an ordinary routine, a reference to the method. So the homogenization theory ceased to be an independent part of mathematical analysis (or the theory of differential equations). The main efforts in homogenization moved from theory to applications in mechanics, physics, biology and so on. Since the problems in periodic structures have been well studied, the attention of researchers was attracted by the more difficult problems in media with special periodicity. These media are given by characteristic functions of the form $\chi^{\varepsilon}(\boldsymbol{x},t)= \chi(\boldsymbol{x},t,\boldsymbol{x}/\varepsilon)$, where $\chi(\boldsymbol{x},t,\boldsymbol{y})$ is a $1$-periodic function of $\boldsymbol{y}$. These problems were investigated in most detail by Meirmanov [15]. He rewrote these models in a special dimensionless form and fixed a certain set of criteria (dimensionless coefficients of differential equations) that are responsible for the type of the physical process (filtration, acoustics, hydraulic shock and so on). Meirmanov classified accurate mathematical models at the microscopic level and used Nguetseng’s multi-scale convergence method to obtain rigorous asymptotic limits of these models. These limits adequately describe the physical processes under consideration at the macroscopic level. In the present paper we take the quantity $\varepsilon=l/L$ for the small dimensionless parameter, where $l$ is the characteristic pore size and $L$ is the characteristic size of the physical domain under consideration. The dimensionless parameter $\alpha_{\mu}^{\varepsilon}$ characterizes the viscosity of the filtered liquid:
$$
\begin{equation*}
\alpha_{\mu}^{\varepsilon}=\frac{2\mu}{Lg\tau\rho_0},
\end{equation*}
\notag
$$
where $\tau$ is the characteristic duration time of the physical process, $\rho_0$ is the density of water, $g$ is the acceleration of gravity, and $\mu$ is the dynamical viscosity of the liquid. We already mentioned that derivation of macroscopic mathematical models must be based on the most accurate mathematical model of the physical process at the microscopic level. This model is described by the laws of the classical continuum mechanics. In our description, these are the Stokes equations in the pore space $\Omega^{\varepsilon}_f$ for an incompressible viscous fluid with dynamical characteristics $\boldsymbol{v}^{\varepsilon}(\boldsymbol{x},t)$ (the velocity vector in the liquid component) and $p^{\varepsilon}(\boldsymbol{x},t)$ (the pressure in the liquid component) and the diffusion-convection equations for the concentration of acid $c^{\varepsilon}(\boldsymbol{x},t)$ and the concentrations of chemical reaction products $c^{\varepsilon}_j(\boldsymbol{x},t)$, $j=1,\dots,n$. We assume that the solid skeleton $\Omega^{\varepsilon}_{s}$ is an absolutely rigid body, that is, the velocities of media in it are equal to zero. The differential equations are supplemented by boundary conditions on the given boundaries, by initial conditions, by the strong discontinuity conditions on the unknown (free) boundary $\Gamma^{\varepsilon}$ ‘solid skeleton–pore space’ in the domain $\Omega=\Omega^{\varepsilon}_f\cup\Gamma^{\varepsilon} \cup \Omega^{\varepsilon}_{s}$ [16] (these conditions arise from the conservation laws of classical mechanics), and by the following condition on free boundary:
$$
\begin{equation}
D^{\varepsilon}_{N}=\alpha^{\varepsilon}c^{\varepsilon},\qquad \alpha^{\varepsilon}=\mathrm{const}>0,
\end{equation}
\tag{1.1}
$$
which is postulated in theoretical chemistry and enables us to find the free boundary [17]. Here $D^{\varepsilon}_{N}$ is the velocity of the boundary $\Gamma^{\varepsilon}$ in the outer normal direction $\boldsymbol{N}^{\varepsilon}$ to $\Gamma^{\varepsilon}$ with respect to the domain $\Omega^{\varepsilon}_f$, and $\alpha$ is a given constant. This statement of the problem appeared in the monograph [18], but no exact results on the existence of a solution were given in that monograph. Note that the physical process under consideration is rather slow (the filtration rate of the liquid is several meters per year). Therefore global-in-time existence theorems for the corresponding initial-boundary value problems are most interesting. On the other hand, since free-boundary problems are strongly non-linear [19], it is usually impossible to prove a global-in-time result for mathematical models at the microscopic level. The only available results are theorems on the existence of a generalized or classical solution of the initial-boundary value problem for the system of differential equations that describes the leaching process at the macroscopic level. But how to obtain a macroscopic mathematical model without any information on the existence of solutions of microscopic mathematical models whose limit is the desired model? To get around these difficulties, we use fixed point theorems [20]. We fix the structure of the pore space given by the characteristic function $\chi(\boldsymbol{x},t,\boldsymbol{y})$. As already mentioned, the arising problem cannot be solved in the case of general position. Therefore it is reasonable to limit ourselves to the simplest cases, for example, the case when the characteristic function $\chi(r,\boldsymbol{y})$ of the pore space is determined by a non-negative function $r(\boldsymbol{x},t)$ belonging to a set $\mathfrak{M}_T$. Then we fix this $r\in \mathfrak{M}_T$ and consider the initial-boundary value problem $\mathbb{B}^{\varepsilon}(r)$ of finding the main characteristics of the medium (the velocity vector, the pressure, and the concentration of acid) in the given domain $\Omega^{\varepsilon}_f(r)$. This problem is the original problem $\mathbb{A}^{\varepsilon}$ without the boundary condition (1.1). In this auxiliary problem with a fixed $\varepsilon>0$, the solid skeleton is a disjoint union of sets of slowly decreasing volume which are close to balls of radius $\varepsilon r$. This simplifies the geometry of the original pore space and enables us to prove the existence of approximate solutions. To understand the structure of the homogenized problem $\mathbb{H}(r)$ of Problems $\mathbb{B}^{\varepsilon}(r)$, we begin with a formal homogenization of Problem $\mathbb{A}^{\varepsilon}$. The conditions for the existence of this homogenization are stated in Lemma 4.1. If $r^{\varepsilon}(\boldsymbol{x},t)$ determines the structure of the solid skeleton (pore space) in Problem $\mathbb{A}^{\varepsilon}$ and if $r_{\varepsilon}\to r^*$ as $\varepsilon\to 0$, the homogenization $\mathbb{H}(r^*)$ of the problem $\mathbb{B}^{\varepsilon}(r^*) =\mathbb{A}^{\varepsilon}$ must coincide with the homogenization $\mathbb{H}$ of Problem $\mathbb{A}^{\varepsilon}$ without homogenization of the boundary condition (1.1). It is clear that homogenization of (1.1) with a given pore space structure $r(\boldsymbol{x},t)$ forms a problem operator whose unique fixed point $r^*(\boldsymbol{x},t)$ determines the required unique homogenization $\mathbb{H}$ of Problem $\mathbb{A}^{\varepsilon}$. To solve Problem $\mathbb{H}(r)$, we have to solve Problem $\mathbb{B}^{\varepsilon}(r)$ and then find its homogenization $\mathbb{H}(r)$ as $\varepsilon\to 0$. The linear problem $\mathbb{B}^{\varepsilon}(r)$ splits into successive solution of the dynamical problem $\mathbb{B}^{\varepsilon}(r)$ of finding the dynamical characteristics $\boldsymbol{v}^{\varepsilon}(\boldsymbol{x},t)$ and $p^{\varepsilon}(\boldsymbol{x},t)$, and the diffusion-convection problem $\mathbb{B}^{\varepsilon}(r)$ of finding the concentration of acid $c^{\varepsilon}(\boldsymbol{x},t)$ and the concentrations of chemical reaction products $c_j^{\varepsilon}(\boldsymbol{x},t)$, $j= 1,\dots,n$. Since these problems are linear, the existence and uniqueness of their generalized solutions follow from the corresponding a priori bounds and known methods of solving linear differential equations, for example, Galerkin’s method [21]. The next step is to homogenize the resulting mathematical model $\mathbb{B}^{\varepsilon}(r)$. To do this, we use Nguetseng’s two-scale convergence method [14]. This enables us to obtain the mathematical model $\mathbb{H}(r)$ in a simple way. But since this method was designed to average only functionals, we need to represent the original mathematical model as a system of integral identities equivalent to the original system of differential equations. The integral identities equivalent to the dynamical Stokes equations are well known, as well as those equivalent to the diffusion-convection equations with classical boundary conditions. But no such identity was known for the concentration of acid, including the diffusion–convection equation and boundary conditions on the free boundary. Representing differential equations in the form of equivalent integral identities is a common and rather difficult task for free-boundary problems. A splendid exception is the Stefan problem [22], [23], which describes phase transitions in pure media (without impurities) such as ‘water–ice’ or chemically pure metals. Oleinik and Kamenomostskaya (the authors of the papers cited) were able to restate the problem in the form of an equivalent integral identity which, in the case when a classical solution exists, contains the heat equation outside the free boundary as well as the condition on the free boundary. This approach enabled them to prove the existence and uniqueness of a weak solution of the integral identity quite simply under minimal requirements on the smoothness of the solution. When a classical solution of the Stefan problem exists, it must coincide with the weak solution. The question of existence of a classical solution of the one-phase Stefan problem was open till 1975 [24]. The existence of a classical solution of the two-phase Stefan problem was proved in 1979 [19]. In our mathematical model of in-situ leaching, it was therefore very important to find an equivalent system of integral identities requiring minimal smoothness of solutions. This was done successfully. It was also necessary to extend the fluid velocity $\boldsymbol{v}^{\varepsilon}$ and the acid concentration $c^{\varepsilon}$, which have been defined only in the pore space $\Omega^{\varepsilon}_{f,T}(r)=\bigcup_{t=0}^{t=T}\Omega^{\varepsilon}_f(r)$, from their domain of definition to the domain $\Omega_T=\Omega\times(0,T)$ preserving their best differential properties. To do this, we used the extension results stated in Lemmas 2.8 and 2.9. The resulting extensions $\widetilde{\boldsymbol{v}}^{\,\varepsilon}(\boldsymbol{x},t)$, $\widetilde{p}^{\,\varepsilon}(\boldsymbol{x},t)$ and $\widetilde{c}^{\,\varepsilon}(\boldsymbol{x},t)$ of the functions $\boldsymbol{v}^{\varepsilon}(\boldsymbol{x},t)$, $p^{\varepsilon}(\boldsymbol{x},t)$ and $c^{\varepsilon}(\boldsymbol{x},t)$ satisfy the integral identities that express the conservation of the momentum, mass and acid concentration and are equivalent to the corresponding differential equations together with the boundary conditions and the initial conditions. The main difficulty in the derivation of a priori bounds for the solutions of $\mathbb{B}^{\varepsilon}(r)$ stems from the moving boundary $\Gamma^{\varepsilon}(r)$. The standard methods of obtaining a priori bounds by multiplying the differential equation by solutions (or their combinations) often do not lead to desired results in the case of free-boundary problems. In our situation, this method helped only partially. In a similar vein, the standard derivation of a priori bounds for solutions of the diffusion problem $\mathbb{B}^{\varepsilon}(r)$ is based on the maximum principle which is inherent in the standard initial-boundary value problems for parabolic equations. Unfortunately, there is no maximum principle for our problem in its original formulation. This motivated us to introduce a modified diffusion problem $\mathbb{B}^{\varepsilon}(r)$ which coincides with the diffusion problem $\mathbb{B}^{\varepsilon}(r)$ in the case when the concentration of acid in the modified diffusion problem $\mathbb{B}^{\varepsilon}(r)$ satisfies the constraints of the standard maximum principle (5.5). A priori bounds for weak solutions (that is, solutions of the corresponding integral identities) are usually obtained by choosing test functions in the integral identity in a special way and integrating by parts. The latter requires sufficient smoothness of the boundary of the pore space $\Omega^{\varepsilon}_f(r)$ (the domain filled with liquid), and this smoothness is determined by the function $r\in\mathfrak{M}_T$. This simple fact is central for the derivation of a priori bounds. We briefly describe the structure of the paper. in § 2 we give well-known facts and definitions and prove new results on the compactness of sequences in a non-periodic structure (Theorem 2.2) and results on the extension of functions defined on the pore space $\Omega^{\varepsilon}_{f,T}(r)$ of a given structure, where the structure is determined by the function $r(\boldsymbol{x},t)$ (Lemmas 2.8 and 2.9). § 3 is devoted to statement of the initial-boundary value problem describing a physical process at the microscopic level. In § 4 we consider formal homogenization of Problems $\mathbb{A}^{\varepsilon}$ and $\mathbb{B}^{\varepsilon}(r)$. Assuming the necessary smoothness of the solutions of the original problem, we obtain integral identities equivalent to the differential equations together with the corresponding boundary conditions and initial conditions. These integral identities enable us to define weak solutions of Problems $\mathbb{A}^{\varepsilon}$ and $\mathbb{B}^{\varepsilon}(r)$. In Lemmas 4.1–12 we state necessary conditions for homogenization of Problems $\mathbb{A}^{\varepsilon}$ and $\mathbb{B}^{\varepsilon}(r)$ and formally derive the homogenized mathematical models $\mathbb{H}(r)$ and $\mathbb{H}$ for the limiting velocity, pressure and acid concentration. In Lemma 4.2, under the assumption (4.18), we derive the integral identity (4.20) equivalent to the boundary condition (1.1) and find the homogenization (4.19) of this boundary condition. Using (4.19), we construct the problem operator
$$
\begin{equation*}
\mathbb{F}(r)=-\theta\int_0^tc(\boldsymbol{x},\tau)\,d\tau
\end{equation*}
\notag
$$
whose fixed points determine the desired structure of the pore space. We also construct auxiliary operators $\mathbb{F}^c(r)=c(\boldsymbol{x},t)$, $\mathbb{F}^v(r)=\boldsymbol{v}(\boldsymbol{x},t)$ and $\mathbb{F}^p(r)=p(\boldsymbol{x},t)$, where $\{\boldsymbol{v},p,c\}$ is a solution of Problem $\mathbb{H}(r)$. The constant $\theta$ is defined by the equality (4.18). In Lemma 12 we prove that the operators $\mathbb{F}^c(r)$, $\mathbb{F}^v(r)$ and $\mathbb{F}^p(r)$ are well defined. In § 5 we state our main results. § 6 is devoted to proving Theorem 5.1. The crucial moment is the derivation of a priori bounds for solutions of Problem $\mathbb{B}^{\varepsilon}(r)$. In § 7 we prove Theorem 5.2 on the existence of weak and classical solutions of Problem $\mathbb{H}(r)$. § 8 is devoted to proving Theorem 5.3. The proof consists in analyzing the properties of the auxiliary operators $\mathbb{F}^c(r)$, $\mathbb{F}^v(r)$, $\mathbb{F}^p(r)$ and the principal operator $\mathbb{F}(r)$. We prove that $\mathbb{F}(r)$ is Lipschitz continuous and its Lipschitz constant is bounded by a linear function of $T$. This enables us to prove the existence of a fixed point $r^*(\boldsymbol{x},t)$ for small values of $T$ and its uniqueness. The last result means the uniqueness of solutions of the mathematical model $\mathbb{H}$. Finally, we use the results on the smoothness of solutions of $\mathbb{H}(r)$ to prove that the time interval $(0,T)$ where the solution of $\mathbb{H}$ exists, can be arbitrary. In this paper we use the notation adopted in [25] and [26]. § 2. Auxiliary assertions2.1. Notation2.1.1. Dimensionless parametersWe have already defined the dimensionless parameters ${\alpha}_{\mu}$ and $c_f$ in § 1. The diffusion of acid and chemical reaction products is characterized by the dimensionless diffusion coefficients
$$
\begin{equation*}
\alpha_{c}=\frac{DT}{L^2}=d_0,\qquad \alpha_i=\frac{D_iT}{L^2},\quad i=1,\dots,n,
\end{equation*}
\notag
$$
for the acid and the chemical reaction products respectively, where $D$ and $D_i$, $i=1,\dots,n$, are the diffusion coefficients. We assume that $\alpha_{\mu}$ depends on a small parameter $\varepsilon$, $\alpha_{\mu}=\alpha^{\varepsilon}_{\mu}$, and there are finite or infinite limits
$$
\begin{equation*}
\lim_{\varepsilon\to 0}\alpha^{\varepsilon}_{\mu}=\mu_0,\qquad \lim_{\varepsilon\to 0} \frac{{\alpha}_{\mu}^{\varepsilon}}{\varepsilon^2}=\mu_1.
\end{equation*}
\notag
$$
The coefficients $c_f$, $d_0$ and $\alpha_i$ are fixed and independent of $\varepsilon$. We write $\varrho_{s}$ (resp. $\varrho_f$) for the dimensionless density of the solid skeleton (resp. of the liquid component) with respect to the density of water $\rho_0$. 2.1.2. Domains and boundariesLet $\Omega\subset \mathbb{R}^3$ be a bounded domain with piecewise smooth boundary ${S}$, $\overline{S}=\partial\Omega$, $S=S^0\cup S^1\cup S^2$, $S^i\cap S^0=\varnothing$, $i=1,2$, $S^1\cap S^2=\varnothing$, $S^j_T=S^j\times(0,T)\subset \mathbb{R}^4$, $j=0,1,2$, $\Omega_T=\Omega\times(0,T)\subset \mathbb{R}^4$. The surface $S^0\subset \mathbb{R}^3$ is impermeable to liquid in the pore space, the surface $S^1\subset \mathbb{R}^3$ simulates injection wells, and the surface $S^2\subset \mathbb{R}^3$ simulates production wells. For simplicity, we shall always assume that $\Omega$ is the unit cube,
$$
\begin{equation*}
\begin{aligned} \, S^0 &=\biggl\{\boldsymbol{x}\colon x_3=\pm\frac{1}{2},\,|x_1|<\frac{1}{2},\,|x_2|<\frac{1}{2} \biggr\} \cup\biggl\{\boldsymbol{x}\colon x_2=\pm\frac{1}{2},\,|x_1|<\frac{1}{2},\,|x_3|<\frac{1}{2}\biggr\}, \\ S^1 &= \biggl\{\boldsymbol{x}\colon x_1=\frac{1}{2},\,|x_2|<\frac{1}{2},\,|x_3|<\frac{1}{2} \biggr\}, \\ S^2 &= \biggl\{\boldsymbol{x}\colon x_1=-\frac{1}{2},\,|x_2|<\frac{1}{2},\,|x_3|<\frac{1}{2} \biggr\}. \end{aligned}
\end{equation*}
\notag
$$
Let $r=r(\boldsymbol{x},t)$, $ 0\leqslant r(\boldsymbol{x},t)\leqslant 1/2$, $(\boldsymbol{x},t)\in \Omega_T$, be the given function that determines the structure of the pore space $\Omega^{\varepsilon}_f(r)\subset\Omega$ and the solid skeleton $\Omega^{\varepsilon}_{s}(r)\subset\Omega$ in $\Omega$ and the structure of the pore space $\Omega^{\varepsilon}_{f,T}(r)=\bigcup_{t=0}^{t=T} \Omega^{\varepsilon}_f(r)\subset \Omega_T$ and the solid skeleton $ \Omega^{\varepsilon}_{s,T}(r)= \bigcup_{t=0}^{t=T}\Omega^{\varepsilon}_{s}(r)\subset \Omega_T$ in $\Omega_T$. We also suppose that the surface $\Gamma^{\varepsilon}(r)= \partial\Omega^{\varepsilon}_f(r) \cap\partial\Omega^{\varepsilon}_{s}(r)$ (resp. $\Gamma_T^{\varepsilon}(r) =\bigcup_{t=0}^{t=T}\Gamma^{\varepsilon}(r)$) divides the liquid and solid components in $\Omega$ (resp. in $\Omega_T$). 2.1.3. The structure of the pore spaceAll functions of the form $\Phi(\boldsymbol{x},t,\boldsymbol{y})$, where $(\boldsymbol{x},t)\in \Omega$ and $\boldsymbol{y}\in \mathbb{R}^3$, are assumed to be $1$-periodic with respect to $\boldsymbol{y}$:
$$
\begin{equation}
\Phi(\boldsymbol{x},t,\boldsymbol{y})= \Phi(\boldsymbol{x},t,\boldsymbol{\varsigma}(\boldsymbol{y})),\qquad \boldsymbol{y}=[|\boldsymbol{y}|]+\boldsymbol{\varsigma}(\boldsymbol{y}),\quad [|\boldsymbol{y}|]=([|y_1|], [|y_2|],[|y_3|]).
\end{equation}
\tag{2.1}
$$
Here $[|a|]$ stands for the integer part of a number $a$. For any fixed $\varepsilon\,{>}\,0$, the pore space $\Omega^{\varepsilon}_f(r)$ and the solid skeleton $\Omega^{\varepsilon}_{s}(r)$ are determined in the following way by a function $\chi(r,\boldsymbol{y})$ which is $1$-periodic with respect to $\boldsymbol{y}$:
$$
\begin{equation}
\begin{gathered} \, \Omega^{\varepsilon}_f(r)=\operatorname{Int}\{\boldsymbol{x}\in\Omega\colon \chi^{\varepsilon}(\boldsymbol{x},t)=1\},\qquad \Omega^{\varepsilon}_{s}(r)=\operatorname{Int}\{\boldsymbol{x}\in\Omega\colon \chi^{\varepsilon}(\boldsymbol{x},t)=0\}, \\ \Omega^{\varepsilon}_f(0)=\varnothing,\qquad \Omega^{\varepsilon}_{s}(0)=\Omega, \\ \chi^{\varepsilon}(\boldsymbol{x},t)=\chi\biggl(r(\boldsymbol{x},t), \frac{\boldsymbol{x}}{\varepsilon}\biggr),\qquad \chi(r,\boldsymbol{y})=\frac{\operatorname{sgn}(r(\boldsymbol{x},t)- |\boldsymbol{\varsigma}(\boldsymbol{y})|)+1}{2}. \end{gathered}
\end{equation}
\tag{2.2}
$$
Consider the decomposition
$$
\begin{equation*}
\begin{gathered} \, \overline{\Omega}=\bigcup_{\boldsymbol{k}\in \mathbb{Z}}\overline{\Omega}^{\,\boldsymbol{k},\varepsilon},\qquad \Omega^{\boldsymbol{k},\varepsilon}=\{\boldsymbol{x}\in\Omega\colon \boldsymbol{x}= \varepsilon\boldsymbol{k}+\varepsilon\boldsymbol{y}\},\quad \Omega_T^{\boldsymbol{k},\varepsilon}= \Omega^{\boldsymbol{k}, \varepsilon}\times(0,T), \\ \Omega_f^{\boldsymbol{k},\varepsilon}(r)= \Omega^{\varepsilon}_f(r)\cap \Omega^{\boldsymbol{k},\varepsilon},\qquad \Omega_{s}^{\boldsymbol{k},\varepsilon}(r)= \Omega^{\varepsilon}_{s}(r)\cap \Omega^{\boldsymbol{k},\varepsilon}, \\ \Omega_{f,T}^{\boldsymbol{k},\varepsilon}(r)= \bigcup_{t=0}^{T}\Omega_f^{\boldsymbol{k},\varepsilon}(r),\qquad \Omega_{s,T}^{\boldsymbol{k},\varepsilon}(r)= \bigcup_{t=0}^{T}\Omega_{s}^{\boldsymbol{k},\varepsilon}(r). \end{gathered}
\end{equation*}
\notag
$$
If $r(\varepsilon\boldsymbol{k},t)=0$, then $\Omega^{\varepsilon}_f(r)=\varnothing$ and $\Omega_{s}^{\boldsymbol{k},\varepsilon}(r)=\Omega^{\boldsymbol{k},\varepsilon}$ for all $\boldsymbol{k}=(k_1,k_2,k_3)$, $k_1,k_2,k_3\in \mathbb{Z}$ (integers) and for all $\boldsymbol{y}=(y_1,y_2,y_3)\in Y$. Here and in what follows, $1/\varepsilon$ is a positive integer. Then $\Omega^{\boldsymbol{k,\varepsilon}}\subset \Omega$ for all $\boldsymbol{k}$. In accordance with this decomposition, the free boundary $\Gamma^{\varepsilon}(r)$ in $\Omega$ is the following set:
$$
\begin{equation*}
\begin{gathered} \, \Gamma^{\varepsilon}(r)=\biggl\{\boldsymbol{x}\in \Omega\colon \biggl| \boldsymbol{\varsigma} \biggl(\frac{\boldsymbol{x}}{\varepsilon}\biggr)\biggr| =r \biggr\} =\bigcup_{\boldsymbol{k}\in \mathbb{Z}}\Gamma^{\boldsymbol{k},\varepsilon}(r)\subset\Omega, \qquad \Gamma^{\varepsilon}_T(r)= \bigcup_{t=0}^{t=T}\Gamma^{\varepsilon}(r) \subset\Omega_T, \\ \Gamma^{\boldsymbol{k},\varepsilon}(r)= \Gamma^{\varepsilon}(r)\cap \Omega^{\boldsymbol{k},\varepsilon},\qquad \Gamma^{\boldsymbol{k},\varepsilon}(r)= \biggl\{\boldsymbol{x}\in \Omega^{\boldsymbol{k},\varepsilon}\colon \biggl|\boldsymbol{\varsigma} \biggl(\frac{\boldsymbol{x}}{\varepsilon}\biggr)\biggr|= r(\boldsymbol{x},t)\biggr\}. \end{gathered}
\end{equation*}
\notag
$$
The structures of the pore space and the solid skeleton are characterized by the elementary cells $Y_f(r)$ and $Y_{s}(r)$:
$$
\begin{equation*}
Y_f(r)=\operatorname{Int}\{\boldsymbol{y}\in Y\colon \chi(r,\boldsymbol{y})=1\},\qquad Y_{s}(r)=\operatorname{Int}\{\boldsymbol{y}\in Y\colon \chi(r,\boldsymbol{y})=0\},
\end{equation*}
\notag
$$
where $Y=(-1/2,1/2)^3\subset \mathbb{R}^3$. Each component $\Gamma^{\varepsilon,\boldsymbol{k}}(r)$ of the surface $\Gamma^{\varepsilon}(r)$ evolves in $\Omega$. We consider the corresponding evolution of the surface $\gamma(r)$ in $Y$, where $\gamma(r)=\{\boldsymbol{y}\in Y\colon |\boldsymbol{y}|=r\}$. In the variables $(\boldsymbol{y},t)$, we write $d_{n}(r)$ for the velocity of $\gamma(r)$ in the direction of the outward normal $\boldsymbol{n}$ to $\gamma(r)$ at $\boldsymbol{y}\in \gamma(r)$ with respect to the domain $Y_f(r)$:
$$
\begin{equation}
d_{n}(r)=-\frac{\partial r}{\partial t}(\boldsymbol{x},t).
\end{equation}
\tag{2.3}
$$
We similarly write $D_{N}^{\varepsilon}(r)$ for the velocity of the free boundary in the direction of the outward normal $\boldsymbol{N}^{\varepsilon}$ to $\Gamma^{\varepsilon}(r)$ at $\boldsymbol{x}\in \Gamma^{\varepsilon}(r)$ with respect to the domain $\Omega^{\varepsilon}_f(r)$. Let $\Psi$ be a function of the form $\Psi=\Psi(\boldsymbol{x},t,\boldsymbol{y})$. We define its averages over the period by putting
$$
\begin{equation*}
\langle\Psi\rangle_{Y}\,{=}\int_{Y}\Psi\,dy,\qquad \langle\Psi\rangle_{Y_f(r)}\,{=}\int_{Y}\chi(r,\boldsymbol{y})\Psi\,dy,\qquad \langle\Psi\rangle_{Y_{s}(r)}\,{=}\int_{Y}(1-\chi(r,\boldsymbol{y}))\Psi\,dy.
\end{equation*}
\notag
$$
In particular, for the given structure $r(\boldsymbol{x},t)$ with characteristic function $\chi(r,\boldsymbol{y})$ defined in (2.1), the function
$$
\begin{equation*}
m(r)=\int_{Y}\chi(r,\boldsymbol{y})\, dy= \langle\chi\rangle_{Y}= 1-\frac{4}{3}\,\pi r^3\geqslant \frac{1}{3},
\end{equation*}
\notag
$$
is the porosity of the solid skeleton at the point $(\boldsymbol{x},t)$. 2.1.4. Differential operators and matricesThe operator $ \nabla$ without any subscripts means differentiation with respect to the variable $\boldsymbol{x}$, and $ \nabla_y$ means differentiation with respect to the variable $\boldsymbol{y}$; $\mathbb{D}(x,\boldsymbol{u})=(1/2)(\nabla_{x}\boldsymbol{u}+ (\nabla_{x}\boldsymbol{u})^*)$; $\mathbb{I}$ is the identity matrix; $\mathbb{B}: \mathbb{C}=\operatorname{tr}(\mathbb{B}\cdot \mathbb{C}^*)$, where $\mathbb{B},\,\mathbb{C}$ are matrices (second-order tensors); the action of a matrix $\mathbb{A}$ on a vector $\boldsymbol{b}$ is denoted by $\mathbb{A}\cdot\boldsymbol{b}$; $\boldsymbol{a}\otimes\boldsymbol{b}$ is a dyad for any vectors $\boldsymbol{a}$, $\boldsymbol{b}$, $\boldsymbol{c}$: $(\boldsymbol{a}\otimes\boldsymbol{b})\cdot \boldsymbol{c}= \boldsymbol{a}(\boldsymbol{b}\cdot\boldsymbol{c})$; $\mathbb{J}^{ij}{=}{(}1/2{)}(\boldsymbol{e}^i{\otimes} \boldsymbol{e}^j{+}\boldsymbol{e}^j{\otimes}\boldsymbol{e}^i)$, where ${\{}{\boldsymbol{e}^1}{,}{\boldsymbol{e}^2}{,}{\boldsymbol{e}^3}{\}}$ is the standard Cartesian basis in $\mathbb{R}^3$; $\mathbb{A}\otimes \mathbb{B}$ is a fourth-order tensor; $(\mathbb{A}\otimes \mathbb{B}):\mathbb{C}=\mathbb{A}(\mathbb{B}:\mathbb{C})$ for any second-order tensors $\mathbb{A}$, $\mathbb{B}$, $\mathbb{C}$. 2.2. Two-scale convergence [14]In this subsection we consider $1$-periodic functions $W(\boldsymbol{x},t,\boldsymbol{y})$ of $\boldsymbol{y}\in Y=(0,1)^3$ with $(\boldsymbol{x},t)\in \Omega_T$. Definition 2.1. We say that a sequence $\{w^{\varepsilon}\}\subset \mathbb{L}_2(\Omega_T)$ is two-scale convergent to a function $W(\boldsymbol{x},t,\boldsymbol{y})\in \mathbb{L}_2(\Omega_T\times Y)$ which is $1$-periodic in the variable $\boldsymbol{y}\in Y$ (and write $w^{\varepsilon}\xrightarrow{\textrm{t.-s.}}W(\boldsymbol{x},t,\boldsymbol{y})$) if the following equality holds for every function $\sigma=\sigma(\boldsymbol{x},t,\boldsymbol{y})$ which is $1$-periodic in $\boldsymbol{y}$: The existence and main properties of two-scale convergent sequences are established in the following theorem. Theorem 2.1 (Nguetseng’s theorem). 1. Every sequence $\{\boldsymbol{w}^{\varepsilon}\}$ which is bounded in $\mathbb{L}_2(\Omega_T)$ contains a subsequence two-scale convergent to some function $\boldsymbol{W}(\boldsymbol{x},t,\boldsymbol{y})$, $\boldsymbol{W}\in\mathbb{L}_2(\Omega_T\times Y)$, $1$-periodic in $\boldsymbol{y}$. 2. Suppose that the sequences $\{\boldsymbol{w}^{\varepsilon}\}$ and $\{\varepsilon\mathbb{D}(x,\boldsymbol{w}^{\varepsilon})\}$ are uniformly bounded in $\mathbb{L}_2(\Omega_T)$. Then there are a function $\boldsymbol{W}=\boldsymbol{W}(\boldsymbol{x},t,\boldsymbol{y})$, $1$-periodic in $\boldsymbol{y}$, and a subsequence of $\{\boldsymbol{w}^{\varepsilon}\}$ (denoted again by $\{\boldsymbol{w}^{\varepsilon}\}$ for simplicity) such that $\boldsymbol{W}, \nabla_{y}\boldsymbol{W}\in \mathbb{L}_2(\Omega_T\times Y)$ and the subsequences $\{\boldsymbol{w}^{\varepsilon}\}$ and $\{\varepsilon\mathbb{D}(x,\boldsymbol{w}^{\varepsilon})\}$ are two-scale convergent in $\mathbb{L}_2(\Omega_T)$ to $\boldsymbol{W}$ and $\mathbb{D}(y,\boldsymbol{W})$ respectively. 3. Suppose that the sequences $\{\boldsymbol{w}^{\varepsilon}\}$ and $\{D(x,\boldsymbol{w}^{\varepsilon})\}$ are bounded in $\mathbb{L}_2(\Omega_T)$. Then one can find functions $\boldsymbol{w}(\boldsymbol{x},t)$, ${\boldsymbol{w}\in \mathbb{W}^{1,0}_2(\Omega_T)}$, and $\boldsymbol{W}(\boldsymbol{x},t,\boldsymbol{y})$, $\boldsymbol{W}\in \mathbb{L}_2(\Omega_T\times Y)\cap\mathbb{W}^{1,0}_2(Y)$, and a subsequence of $\{\mathbb{D}(x,\boldsymbol{w}^{\varepsilon})\}$ such that the function $\boldsymbol{W}$ is 1-periodic in $\boldsymbol{y}$, $\mathbb{D}(x,\boldsymbol{w}) \in \mathbb{L}_2(\Omega_T)$, $D(y,\boldsymbol{W}) \in \mathbb{L}_2(\Omega_T\times Y)$, and the subsequence $\{\mathbb{D}(x,\boldsymbol{w}^{\varepsilon})\}$ is two-scale convergent to the function $\mathbb{D}(x,\boldsymbol{w})+D(y,\boldsymbol{W})$. Corollary 2.1 (see [15], Appendix B, Lemma B.13). If $\alpha(\varepsilon) \|\nabla\boldsymbol{w}^{\varepsilon}\|_{2,\Omega_T}\leqslant M$, where $M$ is independent of $\varepsilon$ and $\lim_{\varepsilon\to 0} (\alpha(\varepsilon)/\varepsilon)=\infty$, then $\boldsymbol{W}(\boldsymbol{x},t,\boldsymbol{y})= \boldsymbol{w}(\boldsymbol{x},t)$. Note that the weak convergence and the two-scale convergence are related by the following implication:
$$
\begin{equation*}
w^{\varepsilon}\xrightarrow{\textrm{t.-s.}} W(\boldsymbol{x},t,\boldsymbol{y}) \quad \Longrightarrow\quad w^{\varepsilon}(\boldsymbol{x})\rightharpoonup \int_Y {W}(\boldsymbol{x},\boldsymbol{y})\, d\boldsymbol{y}\quad (\text{converges weakly}).
\end{equation*}
\notag
$$
2.3. Solenoidal functions on the pore spaceWe put
$$
\begin{equation*}
\mathring{Y}_f\bigl(r(\boldsymbol{x},t)\bigr)= \operatorname{Int}\bigcap_{(\boldsymbol{x},t)\in \Omega_T}Y_f\bigl(r(\boldsymbol{x},t)\bigr).
\end{equation*}
\notag
$$
By construction, $\mathring{Y}_f\neq \varnothing$ because we have $\varnothing\neq Y_f(1/2)\subset Y_f(r)$ for all $r(\boldsymbol{x},t)$, $0\leqslant r(\boldsymbol{x},t)\leqslant 1/2$. Recall that a smooth function $\boldsymbol{\varphi}(\boldsymbol{y})$ is said to be solenoidal in $\mathring{Y}_f$ if $\nabla\cdot \boldsymbol{\varphi}(\boldsymbol{y})=0$ for all $\boldsymbol{y}\in \mathring{Y}_f$. We need the following assertion proved in [15], Appendix B, Lemma B.15. Lemma 2.1. For every unit vector $\boldsymbol{e}$ there is a solenoidal function $\boldsymbol{\varphi}(\boldsymbol{y})$ such that $\boldsymbol{\varphi}\in \mathring{\mathbb{W}}^{1,0}_2(\mathring{Y}_f)$, $\operatorname{supp}\boldsymbol{\varphi}\subset \mathring{Y}_f\subset Y_f(r)$ and 2.4. Strong convergence criteria in $\mathbb{L}_2(\Omega_T)$Definition 2.2. A function $u(\boldsymbol{x},t)$, bounded in $\mathbb{L}_2(\Omega_T)$, possesses temporal derivative $\partial u/\partial t\in\mathbb{L}_2 \bigl((0,T);\mathbb{W}^{-1}_2(\Omega)\bigr)$ if for all functions $\xi\in\mathbb{W}^{1,1}_2(\Omega_T)$, where $C_{u}$ is a positive constant independent of $\xi$. Lemma 2.2 (see [26], [27]). Suppose that the sequences $\{u^{\varepsilon}\}$ and $\{\nabla u^{\varepsilon}\}$ are uniformly bounded in $\mathbb{L}_2(\Omega_T)$ and the sequence of derivatives $\{\partial u^{\varepsilon}\!/\partial t\}$ is uniformly bounded in $\mathbb{L}_2\bigl((0,T);\!\mathbb{W}^{-1}_2(\Omega)\bigr)$. Then there is a subsequence of $\{u^{\varepsilon}\}$ converging strongly in $\mathbb{L}_2(\Omega_T)$. Meirmanov and Zimin generalized this lemma to periodic structures with characteristic function $\chi_0^{\varepsilon}(\boldsymbol{x})\,{=}\, \chi_0(\boldsymbol{x}/\varepsilon)$. Lemma 2.3 (see [28]). Suppose that $\chi_0^{\varepsilon}(\boldsymbol{x})= \chi_0(\boldsymbol{x}/\varepsilon)$, where $\chi_0(\boldsymbol{y})$ is $1$-periodic in $\boldsymbol{y}$, the sequences $\{c^{\varepsilon}\}$ and $\{\nabla c^{\varepsilon}\}$ are uniformly bounded in $\mathbb{L}_2(\Omega_T)$ and the sequence $\{\chi_0^{\varepsilon}(\partial c^{\varepsilon}/\partial t)\}$ is uniformly bounded in $\mathbb{L}_2\bigl((0,T);\mathbb{W}^{-1}_2(\Omega)\bigr)$. Then there is a subsequence of $\{c^{\varepsilon}\}$ converging strongly in $\mathbb{L}_2(\Omega_T)$. Remark 2.1. The norm of an element $\varphi\,{\in} \mathbb{L}_2\bigl((0,T);\mathbb{W}^{-1}_2(\Omega)\bigr)$ is denoted by $\|\varphi\|_{W^{-1}_2}$. In this section we prove a similar result for periodic structures with a special pore space structure. Theorem 2.2. Let the structure $\chi(r,\boldsymbol{y})$ of the pore space be given by (2.1), where $r\in \mathfrak{M}_T$ and Note that the non-positiveness of the temporal derivative of the function $r\in\mathfrak{M}_T$ means that the velocity of the boundary $\Gamma^{\varepsilon}(r)$ in the outward normal direction $\boldsymbol{N}$ to this boundary with respect to the domain $\Omega_f^{\varepsilon}(r)$ is non-positive. We divide the proof of the theorem into several steps. Lemma 2.4. Under the hypotheses of Theorem 2.2, for almost all $t_0\in (0,T)$, the sequence $\{\chi^{\varepsilon}\bigl(r(\boldsymbol{x},t_0)\bigr) \widetilde{c}^{\,\varepsilon}(\boldsymbol{x},t_0)\}$ converges weakly in $\mathbb{L}_2(\Omega)$ to $m(\boldsymbol{x},t_0)\,c(\boldsymbol{x},t_0)$. Proof. By Theorem 2.1, a subsequence of $\{\widetilde{c}^{\,\varepsilon}\}$ is two-scale convergent in $\mathbb{L}_2(\Omega_T)$ to some function $c(\boldsymbol{x},t)$: for all test functions $\varphi\in \mathbb{L}_2(\Omega_T\times Y)$.
By the definition of the temporal derivative of $\widetilde{c}^{\,\varepsilon}$ in $\mathbb{L}_2\bigl((0,T);\mathbb{W}^{-1}_2(\Omega)\bigr)$, there is a sequence of vector-valued functions $\{\boldsymbol{u}^{\varepsilon}\}$ such that $\|\boldsymbol{u}^{\varepsilon}\|^2_{2,\Omega_T}\leqslant M^2$ and
$$
\begin{equation}
\iint_{\Omega_T}\biggl(\widetilde{c}^{\,\varepsilon}(\boldsymbol{x},t) \chi^{\varepsilon}(\boldsymbol{x},t)\,\frac{\partial\varphi}{\partial t}+ \boldsymbol{u}^{\varepsilon}\cdot\nabla\varphi\biggr)\,dx\,dt=0
\end{equation}
\tag{2.7}
$$
for all test functions $\varphi$ vanishing on the lateral boundary of the domain $\Omega_T$ (in $\mathbb{R}^4$).
Here and in what follows we write $M$ for constants depending only on $M_0$ and $T$. Take $\varphi(\boldsymbol{x},t)=\eta(t)\,\psi(\boldsymbol{x})$. Then
$$
\begin{equation}
I=\iint_{\Omega_T}cm(r)\eta\psi\,dx\,dt= \lim_{\varepsilon\to 0} \iint_{\Omega_T}\widetilde{c}\chi^{\varepsilon}(\boldsymbol{x},t) \eta\psi\,dx\,dt.
\end{equation}
\tag{2.8}
$$
We put
$$
\begin{equation*}
f^{\varepsilon}_{\psi}(t)=\int_{\Omega}\chi^{\varepsilon}(\boldsymbol{x},t) \widetilde{c}^{\,\varepsilon}\psi\,dx,\qquad f_{\psi}(t)= \int_{\Omega}m(r)c\psi\, dx.
\end{equation*}
\notag
$$
The limiting relation (2.8) means that
$$
\begin{equation}
I=\lim_{\varepsilon\to 0} \int_0^{T}\eta(t)f^{\varepsilon}_{\psi}(t)\,dt= \int_0^{T}\eta(t)f_{\psi}(t)\, dt.
\end{equation}
\tag{2.9}
$$
Considering again the equality (2.7) with test functions of the form $\xi=\eta(t)\psi(\boldsymbol{x})$, we have
$$
\begin{equation*}
\begin{gathered} \, \int_0^{T}\biggl(\frac{d\eta}{d t} f^{\varepsilon}_{\psi}+\eta U^{\varepsilon}\biggr)\, dt =0,\qquad U^{\varepsilon}(t)=\int_{\Omega}\boldsymbol{u}^{\varepsilon} (\boldsymbol{x},t) \cdot\nabla\psi(\boldsymbol{x})\, dx, \\ \int_0^{T}|U^{\varepsilon}(t)|^2\, dt \leqslant \|\boldsymbol{u}^{\varepsilon}\|^2_{2,\Omega_T} \|\nabla\psi \|^2_{2,Q} \leqslant M^2\|\nabla\psi \|^2_{2,Q}. \end{gathered}
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation*}
\begin{gathered} \, \frac{d f^{\varepsilon}_{\psi}}{d t}(t)=U^{\varepsilon}(t),\qquad f^{\varepsilon}_{\psi}\in \mathbb{W}^1_2(0,T), \\ |f^{\varepsilon}_{\psi}(t)|\leqslant M_{\psi},\qquad |f^{\varepsilon}_{\psi}(t_1)-f^{\varepsilon}_{\psi}(t_2)|\leqslant M_{\psi}\,|t_1-t_2|^{1/2} \end{gathered}
\end{equation*}
\notag
$$
with some constant $M_{\psi}$ independent of $\varepsilon$.
The Arzela–Ascoli theorem ([20], Ch. II, § 7, Theorem 4) enables us to choose a subsequence $\{f^{\varepsilon_{k}}_{\psi}\}$ converging strongly in $\mathbb{C}(\overline{\Omega})$ to some function $\overline{f}_{\psi}(t)$. Letting $\varepsilon_{k}\to 0$ in (2.9) with chosen test functions $\{f^{\varepsilon_{k}}_{\psi}\}$, we arrive at the identity for all test functions $\eta(t)$. Thus $\overline{f}_{\psi}(t)$ coincides with $f_{\psi}(t)$ almost everywhere on $(0,T)$. $\Box$Lemma 2.5. Under the hypotheses of Theorem 2.2, there is a subsequence $\{\varepsilon_{k}\}$ such that for almost all $t_0\in (0,T)$. Proof. It follows from the uniform boundedness of the integrals
$$
\begin{equation*}
\iint_{\Omega_T}|\nabla \widetilde{c}^{\,\varepsilon}(\boldsymbol{x},t)|^2\, dx\, dt
\end{equation*}
\notag
$$
with respect to $\varepsilon_{k}$ that
$$
\begin{equation}
\lim_{\varepsilon_{k}\to 0}\varepsilon_{k}^2\iint_{\Omega_T}|\nabla \widetilde{c}^{\,\varepsilon_{k}}(\boldsymbol{x},t)|^2\, dx\, dt=0.
\end{equation}
\tag{2.11}
$$
We put
$$
\begin{equation*}
u_{k}(t_0)=\varepsilon_{k}^2\int_{\Omega}|\nabla \widetilde{c}^{\,\varepsilon_{k}}(\boldsymbol{x},t_0)|^2\, dx.
\end{equation*}
\notag
$$
Then (2.11) means that the sequence $\{u_{k}\}$ converges to zero strongly in $\mathbb{L}_1(0,T)$.
Using Theorem 1 in [20], Ch. VII, § 1, we conclude that there is a subsequence of $\{u_{k}\}$ converging to zero almost everywhere on $(0,T)$ as $k\,{\to}\,\infty$. $\Box$ Lemma 2.6. Under the hypotheses of Theorem 2.2, the sequence $\{\widetilde{c}^{\,\varepsilon_{k}}(\boldsymbol{x},t_0)\}$ which is bounded in $\mathbb{L}_2(\Omega)$ converges weakly and two-scale in $\mathbb{L}_2(\Omega)$ to the function $c(\boldsymbol{x},t_0)$ on a set of full measure in $(0,T)$. Proof. Choose the sequence $\varepsilon_{k}$ as in Lemma 2.5 and let $\Pi\subset (0,T)$ be a set of full measure on which the conclusion of Lemma 2.5 and the condition (2.10) hold.
Since the sequence $\{\widetilde{c}^{\,\varepsilon_{k}}(\boldsymbol{x},t_0)\}$ is bounded in $\mathbb{L}_2(\Omega)$, it has a subsequence (denoted in the same way for simplicity) that converges two-scale to a function $C(\boldsymbol{x},t_0,\boldsymbol{y})$ which is $1$-periodic in $\boldsymbol{y}$ and belongs to $\mathbb{L}_2(\Omega\times Y)$. Integrating the expression $\varepsilon_{k}\nabla\widetilde{c}^{\,\varepsilon_{k}}(\boldsymbol{x},t_0) \cdot\boldsymbol{\varphi}(\boldsymbol{x}/\varepsilon_{k})\psi(\boldsymbol{x})$ by parts, we arrive at the identity
$$
\begin{equation*}
\begin{aligned} \, &\varepsilon_{k}\int_{\Omega}\nabla\,\widetilde{c}^{\,\varepsilon_{k}} (\boldsymbol{x},t_0)\cdot\boldsymbol{\varphi} \biggl(\frac{\boldsymbol{x}}{\varepsilon_{k}}\biggr)\psi(\boldsymbol{x})\, dx =-\varepsilon_{k}\int_{\Omega}\widetilde{c}^{\,\varepsilon_{k}} (\boldsymbol{x},t_0)\biggl(\boldsymbol{\varphi} \biggl(\frac{\boldsymbol{x}}{\varepsilon_{k}}\biggr)\cdot \nabla\psi(\boldsymbol{x})\biggr)\, dx \\ &\qquad- \int_{\Omega}\widetilde{c}^{\,\varepsilon_{k}}(\boldsymbol{x},t_0) \biggl(\nabla\cdot\boldsymbol{\varphi}\biggl(\frac{\boldsymbol{x}} {\varepsilon_{k}}\biggr)\biggr) \psi(\boldsymbol{x})\, dx, \end{aligned}
\end{equation*}
\notag
$$
which holds for arbitrary functions $\boldsymbol{\varphi}\in\mathbb{W}^1_2(Y)$ and $\psi \in \mathring{\mathbb{W}^1_2}(\Omega)$.
Letting $\varepsilon_{k}\to 0$, we obtain an identity Since $\boldsymbol{\varphi}$ and $\psi$ are arbitrary, this identity is equivalent to the equality $C(\boldsymbol{x},t_0,\boldsymbol{y})=c(\boldsymbol{x},t_0)$. $\Box$Lemma 2.7. Under the hypotheses ofTheorem 2.2, the sequence $\{\widetilde{c}^{\,\varepsilon_{k}}(\boldsymbol{x},t)\}$ converges strongly in $\mathbb{L}_2(\Omega_T)$ to the function $c(\boldsymbol{x},t)$. Proof. To prove this lemma, we put
$$
\begin{equation*}
\begin{gathered} \, \mathbb{H}^1=\mathring{\mathbb{W}^1}_2(\Omega)\subset \mathbb{H}^0 =\mathbb{L}_2(\Omega)\subset \mathbb{H}^{-1}= \mathring{\mathbb{W}^{-1}}_2(\Omega), \\ w_{k}(\boldsymbol{x},t_0)= \widetilde{c}^{\,\varepsilon_{k}}(\boldsymbol{x},t_0) -c(\boldsymbol{x},t_0). \end{gathered}
\end{equation*}
\notag
$$
Using the inequality (9) in [29], Ch. III, § 12, we obtain an inequality
$$
\begin{equation*}
\|w_{k}(\,{\cdot}\,,t_0)\|^2_{\mathbb{H}^0}\leqslant \eta\,\|w_{k}(\,{\cdot}\,,t_0)\|^2_{\mathbb{H}^1}+ C_{\eta}\,\|w_{k}(\,{\cdot}\,,t_0)\|^2_{\mathbb{H}^{-1}}.
\end{equation*}
\notag
$$
Integrate it with respect to time:
$$
\begin{equation}
\int_0^{T}\|w_{k}(\,{\cdot}\,,t)\|^2_{\mathbb{H}^0}\, dt \leqslant \eta\int_0^{T}\|w_{k}(\,{\cdot}\,,t)\|^2_{\mathbb{H}^1}\, dt+ C_{\eta}\int_0^{T}\|w_{k}(\,{\cdot}\,,t)\|^2_{\mathbb{H}^{-1}}\, dt.
\end{equation}
\tag{2.12}
$$
We now use the compactness of the inclusion of $\mathbb{H}^0(\Omega)$ in $\mathbb{H}^{-1}(\Omega)$ (see [30], Ch. 4, § 2, Theorem 3): the weak convergence of the sequence $\{w_{k}(\,{\cdot}\,,t)\}$ in $\mathbb{H}^0(\Omega)$ implies that this sequence converges strongly in $\mathbb{H}^{-1}(\Omega)$.
That is,
$$
\begin{equation*}
\lim_{k\to \infty}\int_0^{T} \|w_{k}(\,{\cdot}\,,t)\|^2_{\mathbb{H}^{-1}}\,dt=0.
\end{equation*}
\notag
$$
Combining this with (2.12), where the positive number $\eta$ can be chosen arbitrarily small, we see that 2.5. Extension lemmasResults on the extension of solutions of differential equations from one domain to another are very important in the theory of homogenization of differential equations (see [10], [11], [31], [32]). For example, when a sequence has different differential properties in different domains and we want to preserve the best differential properties of these solutions under homogenization, the best choice would be to extend every solution from the domain where it has the best differential properties to the domain chosen for homogenization. All currently known results deal with periodic structures. Fortunately, the results of Lemma 1 in [10], Ch. III, § 1 on ‘soft inclusions’ are fully applicable in our case of a non-periodic pore space structure determined by the relations (2.1). This structure is, in principle, a ‘soft inclusion’ and its non-periodicity plays no role since the extension is performed for each individual cell $\Omega^{\boldsymbol{k},\varepsilon}$ of the domain $\Omega$ across the connected component $\Gamma^{\boldsymbol{k},\varepsilon}(r)= \Gamma^{\varepsilon}(r)\cap \Omega^{\boldsymbol{k},\varepsilon}$ of the surface $\Gamma^{\varepsilon}(r)$. Hence the following lemma holds. Lemma 2.8. Suppose that $r\,{\in}\,\mathfrak{M}_T$ and the functions $r(\boldsymbol{x},t)$ determine the structure $\chi(r,\boldsymbol{y})$ of the domain $\Omega_{f,T}^{\varepsilon}$ given by the relations (2.1). Then, for every sequence $\{c^{\varepsilon}\}$ which is bounded in $\mathbb{W}^{1,0}_2(\Omega^{\varepsilon}_{f,T}(r))$, there is an extension $\{\widetilde{c}^{\,\varepsilon}\}$ of the functions $c^{\varepsilon}(\boldsymbol{x},t)$ from $\Omega^{\varepsilon}_{f,T}(r)$ to $\Omega_T$ such that
$$
\begin{equation}
\widetilde{c}^{\,\varepsilon}\in \mathbb{W}^{1,0}_2(\Omega_T),\qquad \|\widetilde{c}^{\,\varepsilon}\|_{2,\Omega_T} \leqslant M\|c^{\varepsilon}\|_{2,\Omega^{\varepsilon}_{f,T}(r)},\quad \|\nabla\widetilde{c}^{\,\varepsilon}\|_{2,\Omega_T} \leqslant M\|\nabla c^{\varepsilon}\|_{2,\Omega^{\varepsilon}_{f,T}(r)}.
\end{equation}
\tag{2.13}
$$
In what follows we shall need functions that assume given values on the connected components $\Gamma^{\boldsymbol{k},\varepsilon}(r)$ of the surface $\Gamma^{\varepsilon}(r)$. Namely, we shall need to extend vector-valued functions $\mathring{\boldsymbol{v}}^{\varepsilon}(\boldsymbol{x},t)$ from the surface $\Gamma^{\varepsilon}(r)$ given by the relations (2.1) in terms of the function $r(\boldsymbol{x},t)$, to the domain $\Omega_T$. At the same time, connected components of the support of this function will be subsets of the set $\Omega^{\boldsymbol{k},\varepsilon}$:
$$
\begin{equation}
\begin{gathered} \, \mathring{\boldsymbol{v}}^{\varepsilon}(r;\boldsymbol{x},t)= \sum_{\boldsymbol{k}\in \mathbb{Z}} \mathring{\boldsymbol{v}}^{\boldsymbol{k},\varepsilon} (r;\boldsymbol{x},t), \qquad \mathring{\boldsymbol{v}}^{\boldsymbol{k},\varepsilon}\in \mathring{\mathbb{W}}^{1,0}_2(\Omega_T^{\boldsymbol{k},\varepsilon}), \\ \|\mathring{\boldsymbol{v}}^{\varepsilon}\|_{2,\Omega_T}+\varepsilon \|\mathbb{D}(x,\mathring{\boldsymbol{v}}^{\varepsilon})\|_{2,\Omega_T} \leqslant \varepsilon MM_0, \\ \nabla\cdot\mathring{\boldsymbol{v}}^{\varepsilon}(r;\boldsymbol{x},t)=0,\qquad (\boldsymbol{x},t)\in \Omega^{\varepsilon}_{f,T},\quad \mathring{\boldsymbol{v}}^{\varepsilon}(r;\boldsymbol{x},t)=0 \ \ \text{for} \ \ \biggl|\boldsymbol{\varsigma} \biggl(\frac{\boldsymbol{x}}{\varepsilon}\biggr)\biggr| \geqslant \frac{5}{12}, \\ \mathring{\boldsymbol{v}}^{\varepsilon}(r;\boldsymbol{x}_0,t)- \bigl(\mathring{\boldsymbol{v}}^{\varepsilon}(r;\boldsymbol{x}_0,t) \cdot \boldsymbol{N}^{\varepsilon}\bigr) \boldsymbol{N}^{\varepsilon}=0, \\ \mathring{\boldsymbol{v}}^{\varepsilon}(r;\boldsymbol{x}_0,t)\cdot \boldsymbol{N}^{\varepsilon}= -\boldsymbol{N}^{\varepsilon}D^{\varepsilon}_{N}((\boldsymbol{x}_0,t)), \qquad \boldsymbol{x}_0\in \Gamma^{\boldsymbol{k},\varepsilon}(r), \end{gathered}
\end{equation}
\tag{2.14}
$$
where $\boldsymbol{N}^{\varepsilon}$ is the outer unit normal (with respect to the domain $\Omega^{\varepsilon}_f(r)$) to the surface $\Gamma^{\varepsilon}(r)$ at the point $\boldsymbol{x}_0$ at time $t$ and $D^{\varepsilon}_{N}$ is the velocity of $\Gamma^{\varepsilon}(r)$ in the direction of the vector $\boldsymbol{N}^{\varepsilon}$. Lemma 2.9. Suppose that $r\in \mathfrak{M}_T$ and the function $r(\boldsymbol{x},t)$ determines the structure $\chi(r,\boldsymbol{y})$ of the domain $\Omega_{f,T}^{\varepsilon}$ given by the relations (2.1). Then, for all $\varepsilon > 0$, there are functions $\mathring{\boldsymbol{v}}^{\varepsilon}(r;\boldsymbol{x},t)$ satisfying the conditions (2.14). Proof. Suppose that $ r^0=\max_{(\boldsymbol{x},t)\in \Omega_T}r(\boldsymbol{x},t)<1/2$ and choose an infinitely smooth periodic function $\boldsymbol{\varsigma}_0(\boldsymbol{y})$ such that
$$
\begin{equation*}
\boldsymbol{\varsigma}_0(\boldsymbol{y})=\begin{cases} \boldsymbol{\varsigma}(\boldsymbol{y}) & \text{for } |\boldsymbol{y}|<r^0, \\ 0 & \text{for } |\boldsymbol{y}|>\dfrac{5}{12}. \end{cases}
\end{equation*}
\notag
$$
Consider a connected component $\Gamma^{\varepsilon,\boldsymbol{k}}(r)$ of $\Gamma^{\varepsilon}(r)$ on which the conditions (2.14) must hold:
$$
\begin{equation}
\biggl|\boldsymbol{\varsigma}_0 \biggl(\frac{\boldsymbol{x}_0}{\varepsilon}\biggr)\biggr|= r(\boldsymbol{x}_0,t),\quad -\frac{1}{\delta}\,\mathring{\boldsymbol{v}}^{\varepsilon} (r;\boldsymbol{x}_0,t)= D^{\varepsilon}_{N}(\boldsymbol{x}_0,t) \boldsymbol{\varsigma}_0 \biggl(\frac{\boldsymbol{x}_0}{\varepsilon}\biggr), \qquad \boldsymbol{x}_0\in \Gamma^{\boldsymbol{k},\varepsilon}(r).
\end{equation}
\tag{2.15}
$$
We look for an extension $\mathring{\boldsymbol{v}}^{\varepsilon}(r;\boldsymbol{x},t)$ of the function $\mathring{\boldsymbol{v}}^{\varepsilon}(r;\boldsymbol{x}_0,t)$ in the form
$$
\begin{equation*}
\mathring{\boldsymbol{v}}^{\varepsilon}(r;\boldsymbol{x},t)= -\delta D^{\varepsilon}_{N}(\boldsymbol{x},t)\boldsymbol{\varsigma}_0 \biggl(\frac{\boldsymbol{x}}{\varepsilon}\biggr) u_{t} \biggl(\boldsymbol{x},\frac{\boldsymbol{x}}{\varepsilon}\biggr),
\end{equation*}
\notag
$$
where $\boldsymbol{x}_0=\varepsilon\boldsymbol{k}+ \varepsilon\boldsymbol{\varsigma}(\boldsymbol{x}_0/\varepsilon)$, $\boldsymbol{x}=\varepsilon\boldsymbol{k}+ \varepsilon\boldsymbol{\varsigma} (\boldsymbol{x}/\varepsilon)$ and the function $u_{t}(\boldsymbol{x},\boldsymbol{x}/\varepsilon)$ will be chosen later. We need to extend the function $\mathring{\boldsymbol{v}}^{\varepsilon}(r;\boldsymbol{x}_0,t)$ from the surface $\Gamma^{\varepsilon,\boldsymbol{k}}(r)$ to a solenoidal function on the ‘liquid component’ of the domain $\Omega^{\boldsymbol{k},\varepsilon}$.
The solenoidality of the extension $\mathring{\boldsymbol{v}}^{\varepsilon}(r;\boldsymbol{x},t)$ is expressed by the following first-order partial differential equation:
$$
\begin{equation}
\begin{aligned} \, 0 &=\!\nabla\cdot\mathring{\boldsymbol{v}}^{\varepsilon}(r;\boldsymbol{x},t) \,{=}\,u_{t}\biggl(\boldsymbol{x},\frac{\boldsymbol{x}}{\varepsilon}\biggr) \biggl(\biggl(\nabla D^{\varepsilon}_{N}(\boldsymbol{x},t)\cdot \boldsymbol{\varsigma}_0 \biggl(\frac{\boldsymbol{x}}{\varepsilon}\biggr)\biggr)+ \frac{1}{\varepsilon}\,D^{\varepsilon}_{N}(\boldsymbol{x},t)\nabla_{y}\cdot \boldsymbol{\varsigma}_0\biggl(\frac{\boldsymbol{x}}{\varepsilon}\biggr)\biggr) \nonumber \\ &\qquad+D^{\varepsilon}_{N}(\boldsymbol{x},t) \biggl(\biggl(\boldsymbol{\varsigma}_0 \biggl(\frac{\boldsymbol{x}}{\varepsilon}\biggr)\cdot \nabla_{x} u_{t} \biggl(\boldsymbol{x},\frac{\boldsymbol{x}}{\varepsilon}\biggr)\biggr)+ \frac{1}{\varepsilon}\biggl(\boldsymbol{\varsigma}_0 \biggl(\frac{\boldsymbol{x}}{\varepsilon}\biggr) \cdot \nabla_{y}u_{t}\biggl(\boldsymbol{x}, \frac{\boldsymbol{x}}{\varepsilon}\biggr)\biggr)\biggr) \end{aligned}
\end{equation}
\tag{2.16}
$$
for the function $u_{t}(\boldsymbol{x},\boldsymbol{x}/\varepsilon)$, together with the initial condition
$$
\begin{equation}
u_{t}\biggl(\boldsymbol{x}_0,\frac{\boldsymbol{x}_0}{\varepsilon}\biggr)=1
\end{equation}
\tag{2.17}
$$
on the surface
$$
\begin{equation}
\Gamma^{\boldsymbol{k},\varepsilon}(r) = \biggl\{ \boldsymbol{x}_0\in\Omega^{\boldsymbol{k},\varepsilon}\colon \boldsymbol{x}_0=\varepsilon\boldsymbol{k}+\boldsymbol{\varsigma} \biggl(\frac{\boldsymbol{x}_0}{\varepsilon}\biggr)\biggr\},
\end{equation}
\tag{2.18}
$$
To simplify the notation, we temporarily omit the superscript $\varepsilon$.
We rewrite the equation (2.16) in the form
$$
\begin{equation}
\boldsymbol{a}_{t}\biggl(\boldsymbol{x}, \frac{\boldsymbol{x}}{\varepsilon}\biggr) \cdot\nabla_{x}\,u_{t}+ \frac{1}{\varepsilon}\,\boldsymbol{a}_{t} \biggl(\boldsymbol{x}, \frac{\boldsymbol{x}}{\varepsilon}\biggr) \cdot\nabla_{y}\,u_{t}= b_{t}\biggl(\boldsymbol{x},\frac{\boldsymbol{x}}{\varepsilon}\biggr) u_{t}\biggl(\boldsymbol{x},\frac{\boldsymbol{x}}{\varepsilon}\biggr),
\end{equation}
\tag{2.19}
$$
where
$$
\begin{equation}
\begin{aligned} \, b_{t}\biggl(\boldsymbol{x},\frac{\boldsymbol{x}}{\varepsilon}\biggr) &= -\biggl(\biggl(\nabla_{x}\,D_{N}(\boldsymbol{x},t) \cdot \boldsymbol{\varsigma}_0 \biggl(\frac{\boldsymbol{x}}{\varepsilon}\biggr)\biggr)+ \frac{1}{\varepsilon}D_{N}(\boldsymbol{x},t) \nabla_{y}\cdot \boldsymbol{\varsigma}_0 \biggl(\frac{\boldsymbol{x}}{\varepsilon}\biggr)\biggr), \nonumber \\ \boldsymbol{a}_{t}\biggl(\boldsymbol{x}, \frac{\boldsymbol{x}}{\varepsilon}\biggr) &= D_{N}(\boldsymbol{x},t)\boldsymbol{\varsigma}_0 \biggl(\frac{\boldsymbol{x}}{\varepsilon}\biggr), \qquad (\boldsymbol{x},t)\in \Omega_T. \end{aligned}
\end{equation}
\tag{2.20}
$$
Our immediate goal is to extend the function $u_{t}(\boldsymbol{x}_0,\boldsymbol{x}_0/\varepsilon)$ from the surface $\Gamma^{\boldsymbol{k},\varepsilon}(r)$ (defined in (2.18)) to the liquid component $\Omega^{\boldsymbol{k},\varepsilon}_f(r)$ of the domain $\Omega^{\boldsymbol{k},\varepsilon}$ along the characteristics of the first-order partial differential equation (2.19).
Let $s$ be the extension parameter and let $\boldsymbol{x}_0\in\Gamma^{\boldsymbol{k},\varepsilon}(r)$ be the starting point of a characteristic of (2.19). By [33], the Cauchy problem (2.17), (2.19) is equivalent to the Cauchy problem for the system of ordinary differential equations
$$
\begin{equation}
\begin{gathered} \, \frac{\partial\boldsymbol{x}}{\partial s}(s,\boldsymbol{x}_0)= \boldsymbol{a}_{t}\biggl(\boldsymbol{x}(s,\boldsymbol{x}_0), \frac{1}{\varepsilon} \, \boldsymbol{x} (s,\boldsymbol{x}_0)\biggr),\qquad \boldsymbol{x}(0,\boldsymbol{x}_0)=\boldsymbol{x}_0, \\ \frac{\partial U_{t}}{\partial s}(s,\boldsymbol{x}_0)= b_{t}\biggl(\boldsymbol{x}(s,\boldsymbol{x}_0), \frac{1}{\varepsilon}\,\boldsymbol{x}(s,\boldsymbol{x}_0)\biggr), \\ U_{t}(0,\boldsymbol{x}_0)=1\quad\text{for}\quad \biggl|\boldsymbol{\varsigma} \biggl(\frac{\boldsymbol{x}_0}{\varepsilon}\biggr)\biggr| = r(\boldsymbol{x}_0,t) \end{gathered}
\end{equation}
\tag{2.21}
$$
for new unknown functions $\boldsymbol{x}(s,\boldsymbol{x}_0)$ and $U_{t}(s,\boldsymbol{x}_0)= u_{t}\bigl(\boldsymbol{x}(s,\boldsymbol{x}_0),(1/\varepsilon) \boldsymbol{x}(s,\boldsymbol{x}_0)\bigr)$.
Since the right-hand sides $\boldsymbol{a}_{t}(\boldsymbol{x},(1/\varepsilon)\boldsymbol{x})$ and $b_{t}(\boldsymbol{x},(1/\varepsilon)\boldsymbol{x})$ are uniformly bounded in $\mathbb{C}^1(\overline{Y}\times \overline{\Omega}_T)$, the Cauchy problem (2.21) has a unique solution $\{\boldsymbol{x}(s,\boldsymbol{x}_0),U_{t}(s,\boldsymbol{x}_0)\}$ for all $(\boldsymbol{x},t)\in \overline{\Omega}_T$ such that $U_{t}(s,\boldsymbol{x}_0)$ and $\partial U_{t}(s,\boldsymbol{x}_0)/\partial s$ depend continuously on $s$ on any bounded interval $s\in [0,S]$ [34] satisfying
$$
\begin{equation}
\biggl|\boldsymbol{\varsigma}_0\biggl(\frac{1}{\varepsilon}\, \boldsymbol{x}(S,\boldsymbol{x}_0) \biggr)\biggr|>0.
\end{equation}
\tag{2.22}
$$
It is natural to ask whether the point $\boldsymbol{x}(s,\boldsymbol{x}_0)$ moves to the solid or liquid component when $s$ increases slightly from its initial position. Since the vector $\boldsymbol{\varsigma}_0(\boldsymbol{x}/\varepsilon)$ at the boundary point is directed to the liquid component, the point $\boldsymbol{x}(s,\boldsymbol{x}_0)$ begins to move into the liquid component: $\boldsymbol{x}(s,\boldsymbol{x}_0)\in \Omega^{\boldsymbol{k},\varepsilon}_f(t)$ for $s>0$.
Differentiating the equality $ U_{t}(s,\boldsymbol{x}_0)= u_{t}\bigl(\boldsymbol{x}(s,\boldsymbol{x}_0),(1/\varepsilon)\boldsymbol{x}(s,\boldsymbol{x}_0) \bigr)$ with respect to $s$, we arrive at another equality:
$$
\begin{equation}
\begin{aligned} \, &\frac{\partial U_{t}}{\partial s}(s,\boldsymbol{x}_0) = \nabla_{x} u_{t}\biggl(\boldsymbol{x}(s,\boldsymbol{x}_0), \frac{1}{\varepsilon}\,\boldsymbol{x}(s,\boldsymbol{x}_0)\biggr) \cdot\frac{\partial \boldsymbol{x}}{\partial s}(s,\boldsymbol{x}_0) \nonumber \\ &\qquad\qquad +\frac{1}{\varepsilon}\,\nabla_{y}u_{t} \biggl(\boldsymbol{x}(s,\boldsymbol{x}_0), \frac{1}{\varepsilon}\,\boldsymbol{x}(s,\boldsymbol{x}_0)\biggr) \cdot \frac{\partial \boldsymbol{x}}{\partial s}(s,\boldsymbol{x}_0) \nonumber \\ &\qquad=\nabla_{x}u_{t}\biggl(\boldsymbol{x}(s,\boldsymbol{x}_0), \frac{1}{\varepsilon}\,\boldsymbol{x}(s,\boldsymbol{x}_0)\biggr) \cdot \boldsymbol{a}_{t} \biggl(\boldsymbol{x}(s,\boldsymbol{x}_0), \frac{1}{\varepsilon}\, \boldsymbol{x}(s,\boldsymbol{x}_0) \biggr) \nonumber \\ &\qquad\qquad+\frac{1}{\varepsilon}\, \nabla_{y}u_{t}\biggl(\boldsymbol{x}(s,\boldsymbol{x}_0), \frac{1}{\varepsilon}\, \boldsymbol{x}(s,\boldsymbol{x}_0)\biggr) \cdot\frac{\partial \boldsymbol{x}}{\partial s}(s,\boldsymbol{x}_0) \cdot\boldsymbol{a}_{t} \biggl(\boldsymbol{x}(s,\boldsymbol{x}_0), \frac{1}{\varepsilon}\,\boldsymbol{x}(s,\boldsymbol{x}_0) \biggr). \end{aligned}
\end{equation}
\tag{2.23}
$$
Thus the second equation in (2.21) and the equation (2.23) prove that (2.16) holds, that is, the function $\mathring{\boldsymbol{v}}^{\varepsilon}(r;\boldsymbol{x},t)$ is solenoidal at all points of $\Omega_f^{\boldsymbol{k},\varepsilon}(r)$ where the condition (2.22) holds. Since we have $\mathring{\boldsymbol{v}}^{\varepsilon}(r;\boldsymbol{x},t)=0$ at any other point, it follows that this function is solenoidal everywhere in the liquid component.
Since, throughout the paper, we use the values of $\mathring{\boldsymbol{v}}^{\varepsilon}(r;\boldsymbol{x},t)$ only on the liquid component, the extension of this function to the solid component is standard. For example, we can extend it along the normal to the boundary and then multiply it by a cut-off function. $\Box$ 2.6. The Leray–Schauder fixed point theoremDefinition 2.3. A continuous operator $\boldsymbol{T}$ from a Banach space $\mathbb{X}$ to itself (notation: $\boldsymbol{T}\colon \mathbb{X} \to \mathbb{X}$) is said to be completely continuous if it maps every closed bounded set to a compact set. Theorem 2.3. Let $\boldsymbol{T}(\lambda, x)$ be a completely continuous operator acting from a bounded subset of a Banach space $\mathbb{X}$ of elements $x\in \mathbb{X}$ to itself and depending continuously on a real parameter $\lambda\in [0,1]$. Suppose that the operator $\boldsymbol{T}(0,x)$ with $\lambda =0$ has a fixed point $x_0$: $\boldsymbol{T}(0,x_0)=x_0$. Then, for every $\lambda\in [0,1]$, the operator $\boldsymbol{T}$ has at least one fixed point $x_{\lambda}$ such that $\boldsymbol{T}(\lambda,x_{\lambda})=x_{\lambda}$. § 3. Statement of the problemLet $r^{\varepsilon}(\boldsymbol{x},t)$ be the function that determines the structure of the liquid and solid components in the domain $\Omega_T$ by the formulae (2.2), where $r_0(\boldsymbol{x})$ is a given function, $ 0\leqslant r^{\varepsilon}(\boldsymbol{x},t) \leqslant 1/2$ and $\widetilde{r}^{\,\varepsilon}(\boldsymbol{x},t)= \max\{0,\, r_0(\boldsymbol{x})-r^{\varepsilon}(\boldsymbol{x},t)\}$. In the dimensionless variables
$$
\begin{equation}
\boldsymbol{x}=\frac{\boldsymbol{x}'}{L},\qquad t=\frac{t'}{\tau},\qquad \boldsymbol{v}=\frac{\tau\boldsymbol{v}'}{L},\qquad p=\frac{p'}{p_{a}},
\end{equation}
\tag{3.1}
$$
where $p_{a}$ is the atmospheric pressure, we look for a solution $\{\widetilde{r}^{\,\varepsilon},\boldsymbol{v}^{\varepsilon},p^{\varepsilon}, c^{\varepsilon},c^{\varepsilon}_j\}$, $j=1,2,\dots,n$, of the following system of differential equations in the unknown domain $\Omega^{\varepsilon}_{f,T} (\widetilde{r}^{\,\varepsilon})$:
$$
\begin{equation}
\nabla\cdot\bigl(\alpha^{\varepsilon}_{\mu}\mathbb{D} (x,\boldsymbol{v}^{\varepsilon})-p^{\varepsilon}\mathbb{I}\bigr)=0,\qquad \nabla\cdot\boldsymbol{v}^{\varepsilon}=0,
\end{equation}
\tag{3.2}
$$
$$
\begin{equation}
\frac{\partial c^{\varepsilon}}{\partial t}=\nabla\cdot (d_0\nabla c^{\varepsilon}-\boldsymbol{v}^{\varepsilon}c^{\varepsilon}),
\end{equation}
\tag{3.3}
$$
$$
\begin{equation}
\frac{\partial c^{\varepsilon}_j}{\partial t}+ \nabla\cdot(\boldsymbol{v}^{\varepsilon}c^{\varepsilon}_j)=0,\qquad j=1,\dots,k.
\end{equation}
\tag{3.4}
$$
This system describes the dynamics of the interaction between the acid solution of concentration $c^{\varepsilon}(\boldsymbol{x},t)$ in the carrier fluid and the solid skeleton $\Omega^{\varepsilon}_{s,T}(\widetilde{r}^{\,\varepsilon})$. In (3.2)–(3.4) we write $\boldsymbol{v}^{\varepsilon}$ (resp. $p^{\varepsilon}$) for the velocity (resp. pressure) of the fluid and $c^{\varepsilon}_j$ for the concentrations of the products of chemical reactions occurring on the free boundary $\Gamma^{\varepsilon}(\widetilde{r}^{\,\varepsilon})= \partial\Omega^{\varepsilon}_f(\widetilde{r}^{\,\varepsilon}) \cap\partial\Omega^{\varepsilon}_{s}(\widetilde{r}^{\,\varepsilon})$. Generally speaking, the dimensionless diffusion coefficient $d_0$ can depend on the concentration of the acid. In our setting, we assume that it is a given positive constant. On the free boundary $\Gamma^{\varepsilon}(\widetilde{r}^{\,\varepsilon})$, the boundary condition (1.1) holds along with the conditions
$$
\begin{equation}
\boldsymbol{v}^{\varepsilon}=v^{\varepsilon}_{N}\boldsymbol{N}^{\varepsilon},
\end{equation}
\tag{3.6}
$$
$$
\begin{equation}
(D^{\varepsilon}_{N}-v^{\varepsilon}_{N})c^{\varepsilon}+ d_0\,\frac{\partial c^{\varepsilon}}{\partial N}= -\beta^{\varepsilon}c^{\varepsilon},
\end{equation}
\tag{3.7}
$$
$$
\begin{equation}
(D^{\varepsilon}_{N}-v^{\varepsilon}_{N}) c^{\varepsilon}_j= -\beta^{\varepsilon}_jc^{\varepsilon},\qquad j=1,\dots,n,
\end{equation}
\tag{3.8}
$$
which express the laws of conservation of mass and momentum [15]–[17]. Here $v^{\varepsilon}_{N}\stackrel{\mathrm{df}}{=} \boldsymbol{v}^{\varepsilon}\cdot\boldsymbol{N}^{\varepsilon}$ is the normal component of the liquid velocity $\boldsymbol{v}^{\varepsilon}$, $\partial c^{\varepsilon}/\partial N\stackrel{\mathrm{df}}{=}\nabla c^{\varepsilon}\cdot\boldsymbol{N}^{\varepsilon}$, $\boldsymbol{N}^{\varepsilon}$ is the outer (with respect to the domain $\Omega^{\varepsilon}_f(\widetilde{r}^{\,\varepsilon})$) unit normal vector to the free boundary $\Gamma^{\varepsilon}(\widetilde{r}^{\,\varepsilon})$, $\delta=(\rho_{s}-\rho_f)/\rho_f$, $\rho_{s}$ (resp. $\rho_f$) is the density of the solid (resp. liquid) component, and $\beta^{\varepsilon}$ and $\beta^{\varepsilon}_j$, $j=1,\dots,n$, are given positive constants. Finally, the following conditions hold on the given boundaries ${S}^1$ (with injective wells) and ${S}^2$ (with producing wells) and on the impenetrable boundary ${S}^0$:
$$
\begin{equation}
\bigl(\alpha^{\varepsilon}_{\mu}\mathbb{D}\bigl(x,\boldsymbol{v}^{\varepsilon} (\boldsymbol{x},t) \bigr)-p^{\varepsilon}(\boldsymbol{x},t)\,\mathbb{I}\bigr) \cdot\boldsymbol{n}(\boldsymbol{x}) =-p^j\boldsymbol{n}(\boldsymbol{x}),\qquad \boldsymbol{x}\in S^j,\quad t>0,\quad j=1,2,
\end{equation}
\tag{3.9}
$$
$$
\begin{equation}
\frac{\partial c^{\varepsilon}}{\partial n}=0,\qquad \boldsymbol{x}\in S^0,\quad t>0,
\end{equation}
\tag{3.10}
$$
$$
\begin{equation}
\boldsymbol{v}^{\varepsilon}=0,\qquad\boldsymbol{x}\in S^0,\quad t>0,
\end{equation}
\tag{3.11}
$$
$$
\begin{equation}
c^{\varepsilon}(\boldsymbol{x},t)=c^0(\boldsymbol{x}),\qquad \boldsymbol{x}\in S^1\cup S^2,\quad t>0,
\end{equation}
\tag{3.12}
$$
$$
\begin{equation}
c^{\varepsilon}_j(\boldsymbol{x},t)=0,\qquad j=1,\dots,n,\quad \boldsymbol{x}\in S^1,\quad t>0.
\end{equation}
\tag{3.13}
$$
In (3.7)–(3.13), $\boldsymbol{n}$ is the unit normal vector to the surfaces $S^0$, $S^1$ and $S^2$. The problem (1.1), (3.1)–(3.13) is completed with the initial conditions
$$
\begin{equation}
c^{\varepsilon}(\boldsymbol{x},0)=c^0(\boldsymbol{x}),\qquad c^{\varepsilon}_j(\boldsymbol{x},0)=0,\quad j=1,\dots,n,\quad \boldsymbol{x}\in \Omega,
\end{equation}
\tag{3.14}
$$
$$
\begin{equation}
\widetilde{r}^{\,\varepsilon}(\boldsymbol{x},0)=0,\qquad \boldsymbol{x}\in \Omega.
\end{equation}
\tag{3.15}
$$
Note that the problem (1.1), (3.2), (3.3), (3.5)–(3.7), (3.9)–(3.12), (3.14), (3.15) of finding the functions $\{\widetilde{r}^{\,\varepsilon}, \boldsymbol{v}^{\varepsilon}, p^{\varepsilon}, c^{\varepsilon}\}$ (we call it Problem $\mathbb{A}^{\varepsilon}$) is independent of the problem of finding the functions $c^{\varepsilon}_j(\boldsymbol{x},t)$, $j=1,2,\dots,n$. The latter problem is to be solved after finding the functions $\{\boldsymbol{v}^{\varepsilon},p^{\varepsilon},c^{\varepsilon}\}$. We assume that $p^i=\mathrm{const}$, $i=1,2$, and $p^0(x_1)= p^1(x_1+1/2)+p^2(x_1-1/2)$. In terms of the new unknown functions $\boldsymbol{v}^{\varepsilon}(\boldsymbol{x},t)$ and $\overline{p}^{\,\varepsilon}(\boldsymbol{x},t)= p^{\varepsilon}(\boldsymbol{x},t)-p^0(x_1)$, the equations (3.2) in the domain $\Omega^{\varepsilon}_{f,T}$ take the form
$$
\begin{equation}
\nabla\cdot\bigl(\alpha^{\varepsilon}_{\mu}\mathbb{D} (x,\boldsymbol{v}^{\varepsilon})-\overline{p}^{\,\varepsilon}\mathbb{I}\bigr)= \nabla\,p^0,\qquad (\boldsymbol{x},t) \in \Omega^{\varepsilon}_{f,T}(\widetilde{r}^{\,\varepsilon}),
\end{equation}
\tag{3.16}
$$
$$
\begin{equation}
\nabla\cdot\boldsymbol{v}^{\varepsilon}=0,\qquad (\boldsymbol{x},t)\in \Omega^{\varepsilon}_{f,T}(\widetilde{r}^{\,\varepsilon}),
\end{equation}
\tag{3.17}
$$
and the boundary condition (3.9) can be rewritten as
$$
\begin{equation}
\bigl(\alpha^{\varepsilon}_{\mu}\mathbb{D}(x,\boldsymbol{v}^{\varepsilon})- \overline{p}^{\,\varepsilon}\mathbb{I}\bigr)\cdot\boldsymbol{N}=0,\qquad \boldsymbol{x}\in S^j,\quad j=1,2,\quad t>0.
\end{equation}
\tag{3.18}
$$
We now transform the equation (3.3) and the boundary condition (3.7) to make the resulting equation and boundary condition equivalent to an integral identity. Namely, we put
$$
\begin{equation}
\frac{\partial c^{\varepsilon}}{\partial t}=\nabla\cdot\biggl(d_0\nabla c^{\varepsilon}-\boldsymbol{v}^{\varepsilon}\biggl(c^{\varepsilon}+ \frac{\beta^{\varepsilon}}{\alpha^{\varepsilon}\delta}\biggr)\biggr), \qquad (\boldsymbol{x},t)\in \Omega^{\varepsilon}_{f,T} (\widetilde{r}^{\,\varepsilon}),
\end{equation}
\tag{3.19}
$$
$$
\begin{equation}
d_0\,\frac{\partial c^{\varepsilon}}{\partial N} -v_{N}^{\varepsilon} \biggl(c^{\varepsilon}+ \frac{\beta^{\varepsilon}}{\alpha^{\varepsilon}\,\delta}\biggr)=0,\qquad (\boldsymbol{x},t)\in\Gamma^{\varepsilon}_T(\widetilde{r}^{\,\varepsilon}).
\end{equation}
\tag{3.20}
$$
The acid concentration $c^{\varepsilon}$ is always non-negative by its original definition, but its non-negativity does not follow automatically from our model since there is no maximum principle for the equation (3.19) and the boundary condition (3.20). Therefore we temporarily replace the equation (3.19) by the equation
$$
\begin{equation}
\frac{\partial c^{\varepsilon}}{\partial t} =\nabla\cdot\bigl(d_0\nabla c^{\varepsilon}-\boldsymbol{v}^{\varepsilon}\psi(c^{\varepsilon})\bigr),\qquad \boldsymbol{x}\in \Omega^{\varepsilon}_f(\widetilde{r}^{\,\varepsilon}),\quad t>0,
\end{equation}
\tag{3.21}
$$
and the boundary condition (3.20) by the boundary condition
$$
\begin{equation}
d_0\,\frac{\partial c^{\varepsilon}}{\partial N}- v^{\varepsilon}_{N} \psi(c^{\varepsilon})= -c^{\varepsilon}D^{\varepsilon}_{N},\qquad \boldsymbol{x}\in\Gamma^{\varepsilon}(\widetilde{r}^{\,\varepsilon}),\quad t>0.
\end{equation}
\tag{3.22}
$$
In (3.22), $\boldsymbol{N}$ is the outer (with respect to the domain $\Omega^{\varepsilon}_f(\widetilde{r}^{\,\varepsilon})$) normal unit vector to $\Gamma^{\varepsilon}(\widetilde{r}^{\,\varepsilon})$ and
$$
\begin{equation}
\psi(s)=s+\frac{\beta^{\varepsilon}}{\alpha^{\varepsilon}\delta}\ \ \text{for}\ \ 0<s<c^*\quad \text{and}\quad\psi(s)=0 \ \ \text{for}\ \ s<0\ \ \text{or for}\ \ s>c^*.
\end{equation}
\tag{3.23}
$$
The equation (3.21) is called the modified diffusion-convection equation for the acid concentration, and the boundary condition (3.22) is called the modified boundary condition for the acid concentration. By Problem $\mathbb{B}^{\varepsilon}(r)$ we mean Problem $\mathbb{A}^{\varepsilon}$ without the boundary condition (1.1), but in a given domain $\Omega^{\,\varepsilon}_{f}(r)$ whose pore space structure $\chi(r,\boldsymbol{y})$ is determined by the function $r(\boldsymbol{x},t)$ in accordance with (2.1). Moreover, the initial-boundary value problem (3.11), (3.16)–(3.18) is referred to as the dynamical problem $\mathbb{B}^{\varepsilon}(r)$, the initial-boundary value problem (3.10), (3.12), (3.14), (3.19), (3.20) is referred to as the diffusion-convection problem $\mathbb{B}^{\varepsilon}(r)$, and the initial-boundary value problem (3.10), (3.12), (3.14), (3.21), (3.22) is referred to as the modified diffusion-convection problem $\mathbb{B}^{\varepsilon}(r)$. § 4. Equivalent integral identities. Formal homogenizationTo state our main results, we need to know the homogenized models $\mathbb{H}$ and $\mathbb{H}(r)$. To obtain the differential equations of macroscopic mathematical models, we must somehow homogenize the microscopic mathematical models $\mathbb{A}^{\varepsilon}$ and $\mathbb{B}^{\varepsilon}(r)$. By the remarks above, it suffices to homogenize only the model $\mathbb{B}^{\varepsilon}(r)$. We assume that
$$
\begin{equation*}
\widetilde{r}^{\,\varepsilon}\in \mathfrak{M},\quad \boldsymbol{v}^{\varepsilon},c^{\varepsilon}\in \mathbb{W}^{1,0}_2\bigl(\Omega^{\varepsilon}_{f,T}(r^{\varepsilon})\bigr) \quad \text{and} \quad \overline{p}^{\,\varepsilon} \in \mathbb{L}_2\bigl(\Omega^{\varepsilon}_{f,T}(r^{\varepsilon})\bigr),\quad |p^0|_{\Omega}^{(2)}=M_0<\infty.
\end{equation*}
\notag
$$
Following the familiar scheme [15], we extend the solutions $\boldsymbol{v}^{\varepsilon}$ and $c^{\varepsilon}$ to $\Omega^{\varepsilon}_{s,T}(r^{\varepsilon})$ and then represent Problem $\mathbb{A}^{\varepsilon}$ in an equivalent form of integral identities such that the conditions on the free boundary are included in the integral identities. By Lemma 2.8, the extension of $c^{\,\varepsilon}$ is the function $\widetilde{c}^{\,\varepsilon}$, and Lemma 2.9 was devoted to the extension of $\boldsymbol{v}^{\,\varepsilon}$. By construction (see Lemma 2.9), the function $\mathring{\boldsymbol{v}}^{\,\varepsilon}$ belongs to $\mathbb{W}^{1,0}_2(\Omega_T)$ and satisfies the boundary conditions (3.4). We put
$$
\begin{equation}
\begin{aligned} \, \widetilde{\boldsymbol{v}}^{\,\varepsilon}(\boldsymbol{x},t) &=\begin{cases} \boldsymbol{v}^{\varepsilon}(\boldsymbol{x},t) &\text{for }(\boldsymbol{x},t) \in \Omega^{\varepsilon}_{f,T}(\widetilde{r}^{\,\varepsilon}), \\ \mathring{\boldsymbol{v}}^{\varepsilon} (\widetilde{r}^{\,\varepsilon};\boldsymbol{x},t) &\text{for }(\boldsymbol{x},t)\in \Omega^{\varepsilon}_{s,T} (\widetilde{r}^{\,\varepsilon}), \end{cases} \\ \widetilde{p}^{\,\varepsilon}(\boldsymbol{x},t) &=\begin{cases} \overline{p}^{\,\varepsilon}(\boldsymbol{x},t) &\text{for }(\boldsymbol{x},t) \in \Omega^{\varepsilon}_{f,T}(\widetilde{r}^{\,\varepsilon}), \\ 0 &\text{for }(\boldsymbol{x},t) \in \Omega^{\varepsilon}_{s,T}(\widetilde{r}^{\,\varepsilon}). \end{cases} \end{aligned}
\end{equation}
\tag{4.1}
$$
Then the function $\widetilde{\boldsymbol{v}}^{\,\varepsilon}\in \mathbb{W}^{1,0}_2(\Omega_T)$ is an extension of $\boldsymbol{v}^{\varepsilon}$ from the domain $\Omega^{\varepsilon}_{f,T}(\widetilde{r}^{\,\varepsilon})$ to $\Omega^{\varepsilon}_{s,T}(\widetilde{r}^{\,\varepsilon})$ satisfying the boundary condition (4.1). Definition 4.1. Suppose that the structure $ \chi^{\varepsilon}(\boldsymbol{x},t)= \chi(r(\boldsymbol{x},t),\boldsymbol{x}/\varepsilon)$ of the pore space $\Omega^{\varepsilon}_{f,T}(r)$ is determined by a function $r\in \mathfrak{M}_T$. We say that the solenoidal function $\widetilde{\boldsymbol{v}}^{\,\varepsilon} \,{\in}\, \mathbb{W}^{1,0}_2(\Omega_T)$ and the functions $\widetilde{p}^{\,\varepsilon}\in \mathbb{L}_2(\Omega_T)$ and $\widetilde{c}^{\,\varepsilon}\in \mathbb{W}^{1,0}_2(\Omega_T)$ constitute a weak solution of Problem $\mathbb{A}^{\varepsilon}$ if they satisfy the integral identities
$$
\begin{equation}
\begin{split} &\iint_{\Omega_{t_0}}\chi^{\varepsilon} \bigl(\alpha^{\varepsilon}_{\mu}\mathbb{D} (x,\widetilde{\boldsymbol{v}}^{\,\varepsilon})- \widetilde{p}^{\,\varepsilon} \mathbb{I}\bigr): \mathbb{D}(x,\boldsymbol{\varphi})\,dx\,dt \\ &\qquad=-\iint_{\Omega_{t_0}}\nabla p^0\cdot\boldsymbol{\varphi}\,dx\,dt - \iint_{\Omega_{t_0}}(1-\chi^{\varepsilon})\sqrt{\alpha^{\varepsilon}_{\mu}}\, \mathring{\mathbb{F}}^{\,\varepsilon}: \mathbb{D}(x,\boldsymbol{\varphi})\,dx\,dt, \end{split}
\end{equation}
\tag{4.2}
$$
$$
\begin{equation}
\iint_{\Omega_{t_0}}\chi^{\varepsilon}\biggl(-\delta\, \frac{\partial\psi}{\partial t}+ \widetilde{\boldsymbol{v}}^{\,\varepsilon} \cdot\nabla\psi\biggr)\,dx\,dt=0
\end{equation}
\tag{4.3}
$$
and
$$
\begin{equation}
\begin{aligned} \, &\int_{\Omega}\chi^{\varepsilon}\widetilde{c}^{\,\varepsilon} (\boldsymbol{x},t_0) \xi(\boldsymbol{x},t_0)\,dx- \int_{\Omega}\chi^{\varepsilon}(\boldsymbol{x},0)\widetilde{c}^{\,\varepsilon} (\boldsymbol{x},0)\xi(\boldsymbol{x},0)\,dx \nonumber \\ &\qquad+\int_0^{t_0}\int_{\Omega}\chi^{\varepsilon} \biggl(-\widetilde{c}^{\,\varepsilon}\, \frac{\partial \xi}{\partial \tau}+ \nabla\xi\cdot\biggl(d_0\nabla\widetilde{c}^{\,\varepsilon}- \widetilde{\boldsymbol{v}}^{\,\varepsilon}\biggl(\widetilde{c}^{\,\varepsilon}+ \frac{\beta^{\varepsilon}}{\alpha^{\varepsilon}\delta}\biggr)\biggr)\biggr)\, dx\, d\tau=0 \end{aligned}
\end{equation}
\tag{4.4}
$$
and the boundary conditions (1.1) and (3.11). The identity (4.2) holds for all smooth functions $\boldsymbol{\varphi}$ vanishing on $S^0$ for $t>0$, as well as for all $t_0$, $0<t_0\leqslant T$. The identity (4.3) holds for all smooth functions $\psi$ vanishing on $S^1\cup S^2$ for $t>0$ and at the initial moment of time, as well as for all $t_0$, $0<t_0\leqslant T$. The identity (4.4) holds for all smooth functions $\xi$ vanishing on the boundary $S^1\cup S^2$ of the domain $\Omega$ for $t>0$, as well as for all $t_0$, $0<t_0\leqslant T$. Remark 4.1. Below, we shall prove that the boundary condition (1.1) is equivalent to the integral identity (4.20). The identity (4.2) is equivalent to the dynamical Stokes equation (3.16). In this identity,
$$
\begin{equation*}
\mathring{\mathbb{F}}^{\varepsilon}=\sqrt{\alpha^{\varepsilon}_{\mu}}\, \mathbb{D}(x,\mathring{\boldsymbol{v}}^{\varepsilon}),\qquad \mathring{\mathbb{F}}^{\varepsilon}\in \mathbb{L}_2(\Omega_T) \quad\text{and}\quad \|\mathring{\mathbb{F}}^{\varepsilon}\|_{2,\Omega_T} \leqslant MM_0,
\end{equation*}
\notag
$$
where the function $\mathring{\boldsymbol{v}}^{\varepsilon}$ is defined by the formula (2.14). The identity (4.3) is equivalent to the solenoidality of $\widetilde{\boldsymbol{v}}^{\,\varepsilon}(\boldsymbol{x},t)$ and the boundary conditions (3.5) and (3.6). The identity (4.4) is equivalent to the diffusion-convection equation (3.21) and the boundary and initial conditions (3.7), (3.10), (3.12), (3.14) and (3.22). The equivalence of these integral identities to the differential equations along with the corresponding boundary conditions follows from the formula of integration by parts (see [16], Part II, § 12) in the form
$$
\begin{equation}
\begin{aligned} \, &\int_0^{t_0}\int_{\Omega_f(t)}\xi \biggl(\frac{\partial A}{\partial t}+\nabla\cdot\boldsymbol{B}\biggr) \, dx\, dt+ \int_0^{t_0}\int_{\Omega_f(t)} \biggl(A\frac{\partial\xi}{\partial t}+ \boldsymbol{B}\cdot\nabla \xi\biggr) \, dx\, dt \nonumber \\ &\qquad=\int_{\Omega_f(t)}\xi(\boldsymbol{x},t_0) A(\boldsymbol{x},t_0)\, dx- \int_{\Omega_f(0)}\xi(\boldsymbol{x},0)A(\boldsymbol{x},0)\, dx \nonumber \\ &\qquad\qquad+\int_0^{t_0}\int_{\Gamma(t)}\xi(\boldsymbol{B}\cdot \boldsymbol{N}- AD_{N})\sin\varphi\,d\sigma\, dt. \end{aligned}
\end{equation}
\tag{4.5}
$$
Here $\boldsymbol{N}\in \mathbb{R}^{3}$ is the unit normal vector to $\Gamma(t)$ pointing outwards $\Omega_{f}(t)$, $D_{N}$ is the normal velocity of the boundary $\Gamma(t)$ in the direction $\boldsymbol{N}$, and $\varphi$ is the angle between the unit vector $\boldsymbol{l}$ of the time axis and the unit normal vector $\boldsymbol{\nu}\in \mathbb{R}^{4}$ to $\Gamma_{T}$ pointing outwards $\Omega_{f,T}$ (hence $\sin\,\varphi=\boldsymbol{\nu}\cdot\boldsymbol{N}$ and $\cos\,\varphi=\boldsymbol{\nu}\cdot \boldsymbol{l}$). For example, in (4.4) we put $A=\widetilde{c}^{\,\varepsilon}$ and $\boldsymbol{B}=\widetilde{\boldsymbol{v}}^{\,\varepsilon} (\widetilde{c}^{\,\varepsilon}+ \beta^{\varepsilon}/ (\alpha^{\varepsilon}\delta))-d_0\nabla\widetilde{c}^{\,\varepsilon}$. Rewriting the diffusion-convection equation (3.3), we arrive at the equalities
$$
\begin{equation*}
\begin{aligned} \, 0 &=\int_0^{t_0}\int_{\Omega}\chi^{\varepsilon}\xi\biggl(\frac{\partial \widetilde{c}^{\,\varepsilon}}{\partial t}+ \nabla\cdot\biggl(\widetilde{\boldsymbol{v}}^{\,\varepsilon} \biggl(\widetilde{c}^{\,\varepsilon}+ \frac{\beta^{\varepsilon}}{\alpha^{\varepsilon}\delta}\biggr)\biggr)- d_0\nabla\widetilde{c}^{\,\varepsilon}\biggr)\, dx\, dt \\ &=\int_0^{t_0}\int_{\Omega}\chi^{\varepsilon}\biggl(\frac{\partial}{\partial t} (\widetilde{c}^{\,\varepsilon}\xi)-\nabla\cdot\biggl(\xi\biggl(d_0\nabla \widetilde{c}^{\,\varepsilon}-\widetilde{\boldsymbol{v}}^{\,\varepsilon} \biggl(\widetilde{c}^{\,\varepsilon}+\frac{\beta^{\varepsilon}} {\alpha^{\varepsilon}\delta}\biggr)\biggr)\biggr)\biggr)\, dx\, dt \\ &\qquad+\int_0^{t_0}\int_{\Omega}\chi^{\varepsilon} \biggl(-\frac{\partial\xi}{\partial t}\, c^{\varepsilon}+\nabla\xi\cdot \biggl(d_0\nabla\widetilde{c}^{\,\varepsilon}- \widetilde{\boldsymbol{v}}^{\,\varepsilon}\biggl(\widetilde{c}^{\,\varepsilon} +\frac{\beta^{\varepsilon}} {\alpha^{\varepsilon}\delta}\biggr)\biggr)\biggr)\,dx\,dt \\ &=\int_0^{t_0}\int_{\Gamma^{\varepsilon}(r)}\xi \biggl(-\widetilde{c}^{\,\varepsilon} D^{\varepsilon}_{N}+ \widetilde{v}^{\,\varepsilon}_{N} \biggl(\widetilde{c}^{\,\varepsilon}+ \frac{\beta^{\varepsilon}}{\alpha^{\varepsilon}\delta}\biggr)- d_0\, \frac{\partial \widetilde{c}^{\,\varepsilon}}{\partial N}\biggr) \sin\varphi\,d \sigma\, dt \\ &\qquad+\int_0^{t_0}\int_{\Omega}\chi^{\varepsilon} \biggl(-\frac{\partial\xi}{\partial t}\, c^{\varepsilon}+\nabla\xi\cdot \biggl(d_0\nabla\widetilde{c}^{\,\varepsilon}- \widetilde{\boldsymbol{v}}^{\,\varepsilon}\biggl(\widetilde{c}^{\,\varepsilon} +\frac{\beta^{\varepsilon}}{\alpha^{\varepsilon}\delta}\biggr)\biggr)\biggr)\, dx\, dt. \end{aligned}
\end{equation*}
\notag
$$
The integrand $\Psi_0=-\widetilde{c}^{\,\varepsilon}D^{\varepsilon}_{N} + \widetilde{v}^{\,\varepsilon}_{N}(\widetilde{c}^{\,\varepsilon}+ \theta/\delta)- d_0(\partial \widetilde{c}^{\,\varepsilon}/\partial N)$ in the integral over the free boundary is equal to zero by the boundary conditions on the free boundary:
$$
\begin{equation*}
D^{\varepsilon}_{N}=\alpha^{\varepsilon}\widetilde{c}^{\,\varepsilon},\qquad \widetilde{v}^{\,\varepsilon}_{N}=-\delta D^{\varepsilon}_{N}=-\delta \alpha^{\varepsilon}\widetilde{c}^{\,\varepsilon},\qquad -d_0\,\frac{\partial \widetilde{c}^{\,\varepsilon}}{\partial N}= \beta^{\varepsilon}\widetilde{c}^{\,\varepsilon} + D^{\varepsilon}_{N}\widetilde{c}^{\,\varepsilon}- \widetilde{v}^{\,\varepsilon}_{N} \widetilde{c}^{\,\varepsilon}.
\end{equation*}
\notag
$$
Definition 4.2. Suppose that the structure $\chi(r,\boldsymbol{y})$ of the pore space $\Omega^{\varepsilon}_{f,T}(r)$ is determined by a function $r\in \mathfrak{M}_T$. Then the functions $\widetilde{\boldsymbol{v}}^{\,\varepsilon}$ and $\widetilde{p}^{\,\varepsilon}$ satisfying the integral identities (4.2) and (4.3) and the boundary condition (3.9), are called a weak solution of the dynamical problem $\mathbb{B}^{\varepsilon}(r)$. Definition 4.3. Suppose that the structure $\chi(r,\boldsymbol{y})$ of the pore space $\Omega^{\varepsilon}_{f,T}(r)$ is determined by a function $r\in \mathfrak{M}_T$, and let $\widetilde{\boldsymbol{v}}^{\,\varepsilon}$ be a weak solution of the dynamical problem $\mathbb{B}^{\varepsilon}(r)$. A function $\widetilde{c}^{\,\varepsilon}{\in}\,\mathbb{V}_2(\Omega_T)$ is called a weak solution of the diffusion-convection problem $\mathbb{B}^{\varepsilon}(r)$ if it satisfies the integral identity (4.4) and the boundary conditions (3.10) and (3.12). Definition 4.4. Suppose that the structure $\chi(r,\boldsymbol{y})$ of the pore space $\Omega^{\varepsilon}_{f,T}(r)$ is determined by a function $r\in \mathfrak{M}_T$, and let $\widetilde{\boldsymbol{v}}^{\,\varepsilon}$ be a weak solution of the dynamical problem $\mathbb{B}^{\varepsilon}(r)$. A function $\displaystyle\,\widetilde{c}^{\,\varepsilon}\in\mathbb{V}_{2}(\Omega_{T})$ is called a weak solution of the modified diffusion-convection problem $\mathbb{B}^{\varepsilon}(r)$ if it satisfies the integral identity
$$
\begin{equation}
\begin{aligned} \, &\int_0^{t_0}\int_{\Omega}\chi^{\varepsilon} \biggl(-\widetilde{c}^{\,\varepsilon}\, \frac{\partial \xi}{\partial t}+\nabla\xi\cdot \bigl(d_0\nabla\widetilde{c}^{\,\varepsilon}- \widetilde{\boldsymbol{v}}^{\,\varepsilon} \psi(\widetilde{c}^{\,\varepsilon})\bigr)\biggr) \, dx\, dt \nonumber \\ &\qquad+\int_{\Omega}\chi^{\varepsilon}(\boldsymbol{x},t_0) \widetilde{c}^{\,\varepsilon} (\boldsymbol{x},t_0) \xi(\boldsymbol{x},t_0)\, dx- \int_{\Omega} \chi^{\varepsilon}(\boldsymbol{x},0) \widetilde{c}^{\,\varepsilon} (\boldsymbol{x},0)\xi(\boldsymbol{x},0)\, dx=0 \end{aligned}
\end{equation}
\tag{4.6}
$$
and the boundary conditions (3.10) and (3.12) for all $t_{0},\,0<t_{0}<T$, and for all smooth functions $\xi$ vanishing on $S^{0}$. In (4.6), the function $\psi(s)$ is given by (3.23). Definition 4.5. Suppose that the structure $\chi^{\varepsilon}(\boldsymbol{x},t)$ of the pore space $\Omega^{\varepsilon}_{f,T}(r)$ is determined by a function $r\in \mathfrak{M}_T$. Let $\widetilde{\boldsymbol{v}}^{\,\varepsilon}$ and $\widetilde{p}^{\,\varepsilon}$ be a weak solution of the dynamical problem $\mathbb{B}^{\varepsilon}(r)$, and let $\widetilde{c}^{\,\varepsilon}$ be a weak solution of the diffusion-convection problem $\mathbb{B}^{\varepsilon}(r)$. Then the triple of functions $\{\widetilde{\boldsymbol{v}}^{\,\varepsilon},\widetilde{p}^{\,\varepsilon}, \widetilde{c}^{\,\varepsilon}\}$ is called a weak solution of Problem $\mathbb{B}^{\varepsilon}(r)$. Lemma 4.1. Suppose that the structure $\chi(r,\boldsymbol{y})$ of the pore space $\Omega^{\varepsilon}_{f,T}(r)$ is determined by a function $r\in \mathfrak{M}_T$ and there is a weak solution $\{\widetilde{\boldsymbol{v}}^{\,\varepsilon},\widetilde{p}^{\,\varepsilon}, \widetilde{c}^{\,\varepsilon}\}$ of Problem $\mathbb{B}^{\varepsilon}(r)$ which is sufficiently smooth for choosing convergent subsequences: Then there are subsequences of the sequences $\{\widetilde{\boldsymbol{v}}^{\,\varepsilon}\}$, $\{\widetilde{p}^{\,\varepsilon}\}$ and $\{\widetilde{c}^{\,\varepsilon}\}$ such that
$$
\begin{equation*}
\begin{gathered} \, \widetilde{\boldsymbol{v}}^{\,\varepsilon}(\boldsymbol{x},t) \rightharpoonup \boldsymbol{v}(\boldsymbol{x},t),\qquad \widetilde{\boldsymbol{v}}^{\,\varepsilon}(\boldsymbol{x},t) \xrightarrow{\textrm{t.-s.}} \boldsymbol{V}(\boldsymbol{x},t,\boldsymbol{y}), \\ \varepsilon\mathbb{D}\bigl(x,\widetilde{\boldsymbol{v}}^{\,\varepsilon} (\boldsymbol{x},t)\bigr) \xrightarrow{\textrm{t.-s.}} \mathbb{D} \bigl(y,\boldsymbol{V}(\boldsymbol{x},t,\boldsymbol{y})\bigr); \\ \widetilde{p}^{\,\varepsilon}(\boldsymbol{x},t) \rightharpoonup p(\boldsymbol{x},t),\qquad \widetilde{p}^{\,\varepsilon}(\boldsymbol{x},t) \xrightarrow{\textrm{t.-s.}} P_f(\boldsymbol{x},t,\boldsymbol{y})\chi(\boldsymbol{x},t,\boldsymbol{y}); \\ \mathring{\boldsymbol{v}}^{\varepsilon}(\boldsymbol{x},t) \quad \textit{strongly converges in } \mathbb{L}_2(\Omega_T)\textit{ to zero}; \\ \mathring{\mathbb{F}}^{\,\varepsilon}(\boldsymbol{x},t)\quad \textit{strongly converges in } \mathbb{L}_2(\Omega_T)\textit{ to zero}; \\ \widetilde{c}^{\,\varepsilon}(\boldsymbol{x},t)\quad \textit{strongly converges in } \mathbb{L}_2(\Omega_T) \textit{ to the function }c(\boldsymbol{x},t); \\ \mathbb{D}(x,\widetilde{c}^{\,\varepsilon}) \xrightarrow{\textrm{t.-s.}} \mathbb{D}(x,c)+ \mathbb{D}\bigl(y,C(\boldsymbol{x},t,\boldsymbol{y})\bigr) \end{gathered}
\end{equation*}
\notag
$$
as $\varepsilon\rightarrow 0$. Here For the limiting procedure, it is necessary to know the behaviour of the coefficients $\alpha^{\varepsilon}_{\mu}$, $\beta^{\varepsilon}$ and $\alpha^{\varepsilon}$ as $\varepsilon\to 0$. In this paper we restrict ourselves to the case of weakly viscous liquid (see [15]) when
$$
\begin{equation}
\mu_0=0,\qquad 0<\mu_1<\infty,\qquad \lim_{\varepsilon\to 0} \frac{\beta^{\varepsilon}}{\alpha^{\varepsilon}}= \theta=\mathrm{const}.
\end{equation}
\tag{4.8}
$$
The passage to the limit in (4.3) is standard (see [15], pp. 6–11):
$$
\begin{equation}
\boldsymbol{v}=-\frac{1}{\mu_1}\,\mathbb{C}^v(r)\cdot\nabla(p+p^0),\qquad (\boldsymbol{x},t)\in \Omega_T,
\end{equation}
\tag{4.9}
$$
$$
\begin{equation}
\nabla\cdot\boldsymbol{v}=\delta\,\frac{\partial\,m}{\partial t},\qquad (\boldsymbol{x},t)\in \Omega_T.
\end{equation}
\tag{4.10}
$$
Moreover, the identities (4.2) and (4.3) contain the boundary conditions
$$
\begin{equation}
v_{n}=\boldsymbol{v}\cdot\boldsymbol{n}=0,\quad\boldsymbol{x}\in S^0,\quad p=0,\quad\boldsymbol{x}\in S^1\cup S^2,\qquad t>0.
\end{equation}
\tag{4.11}
$$
Letting $\varepsilon\to0$ in (4.4), we arrive at an identity
$$
\begin{equation}
\begin{aligned} \, &\int_{\Omega}m(\boldsymbol{x},t_0)c(\boldsymbol{x},t_0) \xi(\boldsymbol{x},t_0)\, dx- \int_{\Omega}m(\boldsymbol{x},0)c^0(\boldsymbol{x})\xi(\boldsymbol{x},0)\, dx \nonumber \\ &\qquad+\int_0^{t_0}\int_{\Omega}\biggl(-mc\,\frac{\partial \xi}{\partial t}+ \nabla\xi\cdot\biggl(d_0\,\mathbb{C}^c(r)\cdot\nabla\,c- \boldsymbol{v}\biggl(c+\frac{\theta}{\delta}\biggr)\biggr)\biggr)\, dx\, d\tau=0, \end{aligned}
\end{equation}
\tag{4.12}
$$
which holds for all smooth functions $\xi$ vanishing on the boundary $S^1\cup S^2$ of the domain $\Omega$ for $t\geqslant 0$. The identity (4.12) is formally equivalent to the homogenized diffusion-convection equation
$$
\begin{equation}
\frac{\partial}{\partial t}(mc)= \nabla\cdot\biggl(d_0\mathbb{C}^c(r)\cdot \nabla c-\boldsymbol{v}\biggl(c+\frac{\theta}{\delta}\biggr)\biggr).
\end{equation}
\tag{4.13}
$$
The symmetric strictly positive definite matrices $\mathbb{C}^v(r)$ and $\mathbb{C}^c(r)$ depend on the structure of the pore space $Y_f(r)$. They are defined by the formulae (1.1.27) and (10.1.61) in [15]:
$$
\begin{equation}
\begin{gathered} \, \mathbb{C}^v(r)=2\int_{Y_f(r)}\sum_{i=1}^3 \bigl(\boldsymbol{V}^i(r;\boldsymbol{y})\otimes\boldsymbol{e}^i\bigr)\,dy, \\ \Delta_{y}\boldsymbol{V}^i-\nabla_{y}\Pi^i=-\boldsymbol{e}^i,\quad \nabla_{y}\cdot\boldsymbol{V}^i=0,\qquad |\boldsymbol{y}|>r, \\ \boldsymbol{V}^i(r;\boldsymbol{y})=0,\qquad |\boldsymbol{y}|=r, \end{gathered}
\end{equation}
\tag{4.14}
$$
$$
\begin{equation}
\begin{gathered} \, \mathbb{C}^c(r)=m\mathbb{I}+\mathbb{C}_0^c(r),\qquad \mathbb{C}_0^c(r)= \int_{Y_f(r)}\biggl(\sum_{i=1}^3\nabla_{y} C^i(r;\boldsymbol{y}) \otimes\boldsymbol{e}^i\biggr)\,dy, \\ \Delta_{y}C^i=0,\qquad |\boldsymbol{y}|>r,\quad \int_{Y_f(r)}C^i(r;\boldsymbol{y})\,dy=0, \\ (\nabla_{y}C^i+\boldsymbol{e}^i)\cdot\boldsymbol{n}=0,\qquad |\boldsymbol{y}|=r, \end{gathered}
\end{equation}
\tag{4.15}
$$
where $\{\boldsymbol{e}^1,\boldsymbol{e}^2,\boldsymbol{e}^3\}$ is the standard orthonormal basis in $\mathbb{R}^3$. The differential equations (4.9), (4.10) and (4.13) are completed with the boundary conditions (4.11) and the boundary and initial conditions
$$
\begin{equation}
\frac{\partial c}{\partial n}=\nabla c\cdot\boldsymbol{n}=0,\qquad \boldsymbol{x}\in S^0, \quad c=c^0,\quad \boldsymbol{x}\in S^1\cup S^2,\quad t>0,
\end{equation}
\tag{4.16}
$$
$$
\begin{equation}
c(\boldsymbol{x},0)=c^0,\qquad \boldsymbol{x}\in \Omega.
\end{equation}
\tag{4.17}
$$
The initial-boundary value problem (4.9)—(4.17) (for a given structure $\chi(\boldsymbol{x},t,\boldsymbol{y})$ of the pore space determined by the function $r\in \mathfrak{M}_{T}$) is referred to as Problem $\mathbb{H}(r)$. The boundary-value problem (4.9)—(4.11) is called the dynamical problem $\mathbb{H}(r)$. Finally, the initial-boundary value problem (4.13), (4.15)–(4.17) is referred to as the diffusion-convection problem $\mathbb{H}(r)$. When no ambiguity is possible, we write $\mathbb{C}^v$ and $\mathbb{C}^c$ instead of $\mathbb{C}^v(r)$ and $\mathbb{C}^c(r)$. In our derivation of Problem $\mathbb{H}(r)$, we did not use the additional boundary condition (1.1) on the free boundary $\Gamma(r)$. Hence Problem $\mathbb{H}(r)$ is a homogenization of Problem $\mathbb{B}^{\varepsilon}(r)$. Thus, to obtain Problem $\mathbb{H}$ as a homogenization of Problem $\mathbb{A}^{\varepsilon}$, we simply complement Problem $\mathbb{H}(r)$ with an equation obtained by homogenizing the boundary condition (1.1) on the free boundary. Then we can find the structure $\chi^{\varepsilon}$ of the pore space by using (2.1). To homogenize the boundary condition (1.1), we need the following assumption. Assumption 4.1. In Problem $\mathbb{A}^{\varepsilon}$ we have where $\theta$ is a given positive constant. Lemma 4.2. Suppose that $\widetilde{c}^{\,\varepsilon}(\boldsymbol{x},t)$ is the extension of $c^{\varepsilon}(\boldsymbol{x},t)$ (see Lemma 2.8), the sequence $\{\widetilde{c}^{\,\varepsilon}\}$ is bounded in $\mathbb{W}^{1,0}_2(\Omega_T)$ and $ \lim_{\varepsilon\to 0}\|\widetilde{c}^{\,\varepsilon}-c\|_{2,\Omega_T}=0$. Then, under Assumption 4.1, the equality
$$
\begin{equation}
d_{n}(\boldsymbol{x},t)=\theta\,c(\boldsymbol{x},t),\qquad \boldsymbol{y}\in \gamma(\boldsymbol{x},t)\subset Y.
\end{equation}
\tag{4.19}
$$
is a homogenization of the boundary condition (1.1). Proof. The boundary condition (1.1) is equivalent to the integral identity
$$
\begin{equation}
\iint_{\Omega_T}\chi^{\varepsilon}\biggl(-\frac{\partial}{\partial t} (\zeta\boldsymbol{a}^{\varepsilon} \cdot \boldsymbol{\xi}_0^{\varepsilon}) +\varepsilon\nabla\cdot (\zeta\,\widetilde{c}^{\,\varepsilon} \boldsymbol{\xi}_0^{\varepsilon})\biggr)\,dx\, dt=0,
\end{equation}
\tag{4.20}
$$
which holds for arbitrary functions $\boldsymbol{\xi}_0^{\varepsilon}=\boldsymbol{\xi}_0 (\boldsymbol{x},t,\boldsymbol{x}/\varepsilon)$ vanishing at $t=0$ and $t=T$, any functions $\zeta(\boldsymbol{x})$ vanishing on the boundary of $\Omega$, and all functions $\boldsymbol{a}^{\varepsilon}(\boldsymbol{x},t) = \boldsymbol{a}(\boldsymbol{x},t,\boldsymbol{x}/\varepsilon)$ such that $\boldsymbol{a}(\boldsymbol{x},t,\boldsymbol{y})= \boldsymbol{n}(\boldsymbol{x},t,\boldsymbol{y})$, where $\boldsymbol{n}(\boldsymbol{x},t,\boldsymbol{y})$ is the outer (with respect to $Y_f(r)$) normal unit vector to $\gamma(\boldsymbol{x},t)$.
Indeed, let $u(\boldsymbol{x},t)$ be an arbitrary function vanishing at $t=0$ and $t=T$. Then the identity (4.5) with $A=1$ and $\boldsymbol{B}=0$ yields that
$$
\begin{equation*}
\int_0^{T}\int_{\Omega_{f,T}^{\varepsilon}(r)} \frac{\partial u}{\partial t}\, dx\, dt=-\iint_{\Gamma^{\varepsilon}_T(r)} D_{N}^{\varepsilon}u\sin \varphi\,d\sigma\,dt.
\end{equation*}
\notag
$$
Here $\varphi$ is the angle between the time axis and the outer (with respect to $\Omega^{\varepsilon}_{f}(r)$) unit normal $\boldsymbol{N}^{\,\varepsilon}$ to the lateral boundary $\Gamma^{\,\varepsilon}(r)$ and $D_{N}^{\varepsilon}$ is the velocity of $\Gamma^{\varepsilon}(r;t)$ in the direction $\boldsymbol{N}^{\varepsilon}$.
In the integral over $\Gamma_T^{\varepsilon}(r)$, we use the boundary condition (1.1) and Assumption 4.1:
$$
\begin{equation*}
\iint_{\Omega_T}\chi^{\varepsilon}\, \frac{\partial u}{\partial t}\, dx\, dt= -\iint_{\Gamma^{\varepsilon}_T(r)} \alpha^{\varepsilon}\widetilde{c}^{\,\varepsilon} u\sin \varphi\,d\sigma\,dt= -\iint_{\Gamma^{\varepsilon}_T(r)}\varepsilon\theta \widetilde{c}^{\,\varepsilon} u\sin \varphi\,d\sigma dt.
\end{equation*}
\notag
$$
We now put
$$
\begin{equation*}
u=\boldsymbol{a}^{\varepsilon}(\boldsymbol{x},t)\cdot \boldsymbol{\xi}^{\varepsilon}(\boldsymbol{x},t),\qquad \boldsymbol{a}^{\varepsilon}(\boldsymbol{x},t)= \boldsymbol{a}\biggl(\boldsymbol{x},t, \frac{\boldsymbol{x}}{\varepsilon}\biggr),\quad \boldsymbol{\xi}^{\varepsilon}(\boldsymbol{x},t)= \zeta(\boldsymbol{x})\boldsymbol{\xi}_0 \biggl(\frac{\boldsymbol{x}}{\varepsilon},t\biggr),
\end{equation*}
\notag
$$
where the functions $\boldsymbol{a}(\boldsymbol{x},t,\boldsymbol{y})$ and $\boldsymbol{\xi}_0(\boldsymbol{y},t)$ are $1$-periodic in $\boldsymbol{y}$ and $\boldsymbol{a}(\boldsymbol{x},t,\boldsymbol{y})= \boldsymbol{n}(\boldsymbol{x},t,\boldsymbol{y})$ for $\boldsymbol{y}\in \gamma(\boldsymbol{x},t)$. We have
$$
\begin{equation*}
\begin{aligned} \, &\iint_{\Omega_T}\chi^{\varepsilon}\, \frac{\partial}{\partial t} (\boldsymbol{a}^{\varepsilon}\cdot\boldsymbol{\xi}^{\varepsilon})\, dx\, dt+ \int_0^{T}\int_{\Gamma^{\varepsilon}(r;t)}\varepsilon\theta \widetilde{c}^{\,\varepsilon}(\boldsymbol{n}\cdot \boldsymbol{\xi}^{\varepsilon}) \sin\varphi\, d\sigma\, dt \\ &\qquad=\iint_{\Omega_T}\chi^{\varepsilon}\biggl(\frac{\partial}{\partial t} (\zeta\boldsymbol{a}^{\varepsilon}\cdot\boldsymbol{\xi}_0^{\varepsilon})+ \varepsilon\theta\nabla\cdot(\zeta\,\widetilde{c}^{\,\varepsilon} \boldsymbol{\xi}_0^{\varepsilon})\biggr)\, dx\, dt=0. \end{aligned}
\end{equation*}
\notag
$$
The resulting identity
$$
\begin{equation}
\iint_{\Omega_T}\chi^{\varepsilon} \biggl(\frac{\partial}{\partial t}(\zeta\boldsymbol{a}^{\varepsilon} \cdot\boldsymbol{\xi}_0^{\varepsilon})+ \varepsilon\theta\nabla\cdot(\zeta\,\widetilde{c}^{\,\varepsilon} \boldsymbol{\xi}_0^{\varepsilon})\biggr)\, dx\, dt=0
\end{equation}
\tag{4.21}
$$
is equivalent to the boundary condition (1.1).
It holds for all functions $\boldsymbol{a}^{\varepsilon}(\boldsymbol{x},t)= \boldsymbol{a}(\boldsymbol{x},t,\boldsymbol{x}/\varepsilon)$ such that $\boldsymbol{a}(\boldsymbol{x},t,\boldsymbol{y})= \boldsymbol{n}(\boldsymbol{x},t,\boldsymbol{y})$, where $\boldsymbol{n}(\boldsymbol{x},t,\boldsymbol{y})$ is the outer (with respect to $Y_{r,f}(\boldsymbol{x},t)$) unit normal vector to $\gamma(\boldsymbol{x},t)$, for arbitrary functions $\boldsymbol{\xi}_0(\boldsymbol{y},t)$ vanishing at $t=0$ and $t=T$, and for arbitrary functions $\zeta(\boldsymbol{x})$ vanishing on the boundary $\partial\Omega$. Letting $\varepsilon\to 0$ in (4.21), we obtain
$$
\begin{equation*}
\int_{\Omega}dx\,\zeta\,\int_0^{T}\int_{Y_f(r)}\biggl(\frac{\partial}{\partial t} (\boldsymbol{a}\cdot\boldsymbol{\xi}_0)+ \nabla_{y}\cdot(\theta c \boldsymbol{\xi}_0) \biggr) \,dy\, dt=0.
\end{equation*}
\notag
$$
Here we have used the last part of Theorem 2.1 on the two-scale convergence of functions in $\mathbb{W}^{1,0}_2(\Omega_T)$ to a function independent of the rapid variable $\boldsymbol{y}$.
The desired equality (4.19) follows from the last identity for arbitrary functions $\zeta$ and $\boldsymbol{\xi}_0$ and from the Gauss–Ostrogradskii theorem (4.5):
$$
\begin{equation*}
\begin{aligned} \, 0 &= \int_0^{T}\int_{Y_f(r)}\biggl(\frac{\partial}{\partial t}(\boldsymbol{a}\cdot\boldsymbol{\xi}_0)+ \nabla_{y}\cdot (c\,\boldsymbol{\xi}_0) \biggr) \, dy\, dt \\ &=\int_0^{T}\int_{\gamma}(-d_{n}+\theta c) (\boldsymbol{n}\cdot \boldsymbol{\xi}_0) \sin\varphi \,d\sigma\, dt=0. \end{aligned}
\end{equation*}
\notag
$$
The boundary condition (4.19) implies that
$$
\begin{equation*}
\frac{\partial r}{\partial t}(\boldsymbol{x},t)=-\theta c(\boldsymbol{x},t).
\end{equation*}
\notag
$$
This suggests a scheme for proving the existence of a solution of Problem $\mathbb{H}$ using the Schauder fixed point theorem.
Let $\mathfrak{M}_{T}$ be the set of functions $r(\boldsymbol{x},t)$ that determine the structure of the pore space $\Omega^{\varepsilon}_{f}(r)$ in the variables $(\boldsymbol{x},t)$ and the pore space $\displaystyle Y_{f}(r)$ in the variables $(\boldsymbol{y},t)$. Given the structure of the pore space, we find a solution $\{\boldsymbol{v}\,{=}\,\mathbb{F}^v(r), \nabla p= \mathbb{F}^p(r),\, c=\mathbb{F}^c(r)\}$ of Problem $\mathbb{H}(r)$ and the function
$$
\begin{equation}
R(\boldsymbol{x},t)=r_0(\boldsymbol{x})- \theta \int_0^{t_0}c(\boldsymbol{x},\tau)\, d\tau \equiv \mathbb{F}(r),
\end{equation}
\tag{4.22}
$$
which determines a new structure of the pore space by the formulae (2.2).
Fixed points $r_*(\boldsymbol{x},t)$ of the operator $\mathbb{F}(r)$ determine (by (2.2)) the characteristic function $ \chi(r_*,\boldsymbol{y})$ of the pore space $ Y_f(r_*)$ for which Problem $\mathbb{H}(r_*)$ coincides with Problem $\mathbb{H}$. $\Box$ Lemma 12. Suppose that Then the operators $\mathbb{F}^c(r)$, $\mathbb{F}^v(r)$ and $\mathbb{F}^p(r)$ are well defined. Proof. The difference $\widetilde{p}=p_1-p_2$ of two possible weak solutions $p_1$ and $p_2$ of the problem (4.9)–(4.11) satisfies the homogeneous elliptic equation
$$
\begin{equation}
\nabla\cdot(\mathbb{C}^v\cdot \nabla\widetilde{p}\,)=0
\end{equation}
\tag{4.23}
$$
in the domain $\Omega_T$ with homogeneous boundary conditions (4.11) on the boundary $S$ of the domain $\Omega$ for all $t>0$.
Since the matrix $\mathbb{C}^v$ is strictly positive definite, the solution of the problem (4.11), (4.23) is unique. Hence the operators $\mathbb{F}^v(r)$ and $\mathbb{F}^p(r)$ are well defined. Consider the difference $u=m_1c_1-m_2c_2$, $m_i=m(r_i)$, $i=1,2$, of two possible solutions $c_1$ and $c_2$ of the problem (4.13), (4.16), (4.17). It satisfies the difference of the integral identities (4.4):
$$
\begin{equation*}
\begin{aligned} \, &\int_{\Omega}u(\boldsymbol{x},t_0)\,\xi(\boldsymbol{x},t_0)\, dx+ \int_0^{t_0}\int_{\Omega}\biggl(-u\,\frac{\partial \xi}{\partial t}+ \frac{d_0}{m}\nabla\xi\cdot \mathbb{C}^c\cdot\nabla u\biggr)\, dx\, dt \\ &\qquad=\int_0^{t_0}\int_{\Omega}\nabla\xi\cdot\biggl(\boldsymbol{v}\, \frac{u}{m}+\frac{u}{m^2}B^c\cdot\nabla m\biggr)\, dx\, dt, \end{aligned}
\end{equation*}
\notag
$$
where $\xi$ is an arbitrary smooth function vanishing for $\boldsymbol{x}\,{\in}\, S^1\cup S^2$ and $0\,{<}\,t\,{<}\,T$, along with the following homogeneous boundary and initial conditions:
$$
\begin{equation*}
\begin{gathered} \, u(\boldsymbol{x},t)=0,\qquad\boldsymbol{x}\in S^1\cup S^2,\quad 0<t<T, \\ \frac{\partial u}{\partial N}=0,\qquad\boldsymbol{x}\in S^0,\quad 0<t<T,\quad u(\boldsymbol{x},0)=0,\quad\boldsymbol{x}\in \Omega. \end{gathered}
\end{equation*}
\notag
$$
Using the standard procedure of smoothing in time (see [25], Ch. II, § 4) with test function $\xi=(u_{h})_{\bar{h}}$ and the inequality $ab\leqslant \lambda a^2 +C_{\lambda}b^2$, we see that
$$
\begin{equation*}
\begin{aligned} \, &\int_{\Omega}(u_{h})_{\bar{h}}(\boldsymbol{x},t)u(\boldsymbol{x},t)\, dx- \frac{1}{2}\int_{\Omega}u^2_{h}(\boldsymbol{x},t)\,dx \\ &\qquad+\iint_{\Omega_{t}}\frac{d_0}{m}(u_{h})_{\bar{h}}\cdot(\mathbb{C}^c \cdot\nabla u) \, dx\, dt \leqslant \lambda\iint_{\Omega_{t}}|(\nabla u)_{\bar{h}}|^2\, dx\, d\tau + C_{\lambda}\iint_{\Omega_{t}}u^2\, dx\, d\tau. \end{aligned}
\end{equation*}
\notag
$$
Letting $h\to 0$, using the strict positive definiteness of the matrix $\mathbb{C}^c$:
$$
\begin{equation*}
\frac{d_0}{m}\boldsymbol{\xi}\bigl(\,{\cdot}\, (\mathbb{C}^c\cdot\boldsymbol{\xi})\bigr)\geqslant \nu_0 |\boldsymbol{\xi}|^2,
\end{equation*}
\notag
$$
and choosing sufficiently small $\lambda$, we obtain a differential inequality
$$
\begin{equation*}
\int_{\Omega}u^2(\boldsymbol{x},t)\,dx \leqslant M^2\iint_{\Omega_{t}}u^2(\boldsymbol{x},t)\,dx \, d\tau,\qquad u(\boldsymbol{x},0)=0.
\end{equation*}
\notag
$$
By Gronwall ’s lemma (see [25], Ch. II, § 5, Lemma 5.5), it follows that $c(\boldsymbol{x},t)=0$ almost everywhere in $\Omega_T$. Hence the operator $\mathbb{F}^c(r)$ is well defined. $\Box$ § 5. The main resultTheorem 5.1. Suppose that $c^0\in \mathbb{C}^{2+\gamma}(\overline{\Omega})$, the conditions (4.8) hold and Then, for every $r\in \mathfrak{M}_T$, Problem $\mathbb{B}^{\varepsilon}(r)$ has a unique weak solution $\{\boldsymbol{v}^{\varepsilon},\overline{p}^{\,\varepsilon},c^{\varepsilon}\}$ satisfying the estimates
$$
\begin{equation}
\|\overline{p}^{\,\varepsilon}\|_{2,\Omega^{\varepsilon}_{f,T}(r)}+ \|\boldsymbol{v}^{\varepsilon}\|_{2,\Omega^{\varepsilon}_{f,T}(r)}+ \|\sqrt{\alpha_{\mu}}\, \mathbb{D}(x,\boldsymbol{v}^{\varepsilon})\|_{2,\Omega^{\varepsilon}_{f,T}(r)} \leqslant M(M_0)|\nabla p^0|^{(0)}_{\Omega}=M_{p},
\end{equation}
\tag{5.2}
$$
$$
\begin{equation}
\|c^{\varepsilon}\|_{2,\Omega^{\varepsilon}_{f,T}(r)}+ \|\nabla c^{\varepsilon}\|_{2,\Omega^{\varepsilon}_{f,T}(r)} \leqslant M(M_0) (|c^0|^{(2+\gamma)}_{\Omega}+M_{p})=M_{c},
\end{equation}
\tag{5.3}
$$
where the constants $M(M_{p})$ and $M(M_{c})$ are independent of $\varepsilon$. Theorem 5.2. Under the hypotheses of Theorem 5.1, suppose that Assumption 4.1 holds. Then, for every $T>0$, Problem $\mathbb{H}(r)$ has a unique classical solution $\{\boldsymbol{v},p,c\}$ such that $\nabla p,\boldsymbol{v} \in\mathbb{H}^{\gamma,\gamma/2}(\overline{\Omega}_T)$, $c\in \mathbb{H}^{2+\gamma,\, (2+\gamma)/2}(\overline{\Omega}_T)$,
$$
\begin{equation}
|\nabla p|_{\Omega_T}^{(\gamma)}+ |\boldsymbol{v}|_{\Omega_T}^{(\gamma)}\leqslant M(M_{p}),
\end{equation}
\tag{5.6}
$$
Theorem 5.3. Under the hypotheses of Theorem 5.2, for every $T>0$, Problem $\mathbb{H}$ has a unique classical solution $\{r^*,\boldsymbol{v},p,c\}$ such that $r^*\in \mathfrak{M}_T$, $\nabla p,\boldsymbol{v} \in\mathbb{H}^{\gamma,\gamma/2}(\overline{\Omega}_T)$, $c\in \mathbb{H}^{2+\gamma,\, (2+\gamma)/2}(\overline{\Omega}_T)$ and the estimates (5.6)–(5.8) hold. § 6. Proof of Theorem 5.1By [25], Ch. IV, § 9 (well-posedness of the diffusion-convection problem) and [35], Ch. III, § 5, Theorem 2 (well-posedness of the dynamical problem), we have
$$
\begin{equation*}
c^{\varepsilon}\in \mathbb{H}^{2+\gamma,\, (2+\gamma)/2} \bigl(\overline{\Omega}^{\,\varepsilon}_{f,T}(r)\bigr),\quad \boldsymbol{v}^{\varepsilon}, \nabla p^{\varepsilon}\in \mathbb{L}_{\infty}\bigl(0,T;\mathbb{W}^2_{q} \bigl(\Omega^{\varepsilon}_f(r)\bigr)\bigr)\cap\mathbb{H}^{\gamma,\gamma/2} \bigl(\overline{\Omega}^{\,\varepsilon}_{f,T}(r)\bigr),
\end{equation*}
\notag
$$
where $q>3(1+\gamma)$ is arbitrary. Note that these theorems were stated for cylindrical domains. However, using the local estimates, we can reprove them for non-cylindrical domains. The same argument as in Lemma 7.5 (the estimate (7.25)) proves that the function $\boldsymbol{v}^{\varepsilon}$ belongs to the space$\mathbb{H}^{\gamma,\gamma/2} \bigl(\overline{\Omega}^{\,\varepsilon}_{f,T}(r)\bigr)$. The dynamical problem $\mathbb{B}^{\varepsilon}(r)$ is linear. Therefore, to prove the existence of weak solutions of Problem $\mathbb{B}^{\varepsilon}(r)$, it suffices to obtain appropriate a priori estimates for solutions of these problems. 6.1. A priori bounds for weak solutions of the dynamical problem $\mathbb{B}^{\varepsilon}(r)$For simplicity we shall always omit the superscript $\varepsilon$ if this does not lead to ambiguities. Moreover, the domains $\Omega^{\varepsilon}_f\bigl(r(\boldsymbol{x},t)\bigr)$, $\Omega^{\varepsilon}_{s}\bigl(r(\boldsymbol{x},t)\bigr)$, $\Omega^{\boldsymbol{k},\varepsilon}_f\bigl(r(\boldsymbol{x},t)\bigr)$ and the boundary $\Gamma\bigl(r(\boldsymbol{x},t)\bigr)$ will be denoted by $\Omega_f(t)$, $\Omega_{s}(t)$, $\Omega^{\boldsymbol{k}}_f(t)$ and $\Gamma(t)$. Lemma 6.1. Suppose that $r\in \mathfrak{M}_T$. Then, under the hypotheses of Theorem 5.1, the estimates (5.2) hold for all weak solutions of the dynamical problem $\mathbb{B}(r)$. Proof. Taking $\widetilde{\boldsymbol{v}}$ for the test function $\boldsymbol{\varphi}$ in the integral identity (4.2), we have
$$
\begin{equation*}
\begin{aligned} \, \int_{\Omega_f(t)}\alpha_{\mu} |\mathbb{D}(x,\widetilde{\boldsymbol{v}})|^2\, dx &= -\int_{\Omega_f(r)}(1-\chi)\sqrt{\alpha_{\mu}}\, \mathbb{D}(x,\widetilde{\boldsymbol{v}}): \mathring{\mathbb{F}}\,dx \\ &\leqslant\nu\int_{\Omega_f(t)}\alpha_{\mu} |\mathbb{D}(x,\widetilde{\boldsymbol{v}})|^2\, dx+ \frac{4}{\nu}\int_{\Omega_{s}(t)}|\mathring{\mathbb{F}}|^2\, dx, \end{aligned}
\end{equation*}
\notag
$$
where $\mathring{\mathbb{F}}=\chi\sqrt{\alpha_{\mu}}\, \mathbb{D}(x,\mathring{\boldsymbol{v}})$ and $\nu$ is an arbitrarily small positive number.
Putting $\nu=\mu_1/4$ and using the uniform boundedness with respect to $\varepsilon$ of the functions $\mathring{\mathbb{F}}$ in the space $\mathbb{L}_2(\Omega_{s,T})$ (the estimate (2.14)), we obtain the required estimate for the summand $\sqrt{\alpha_{\mu}}\,\mathbb{D}(x,\widetilde{\boldsymbol{v}})$:
$$
\begin{equation}
\int_{\Omega_f(t)}\alpha_{\mu} \mathbb{D}(x,\widetilde{\boldsymbol{v}})|^2\, dx \leqslant M(M_0)|\nabla\,p^0|^{(0)}_{\Omega}=M_{p}.
\end{equation}
\tag{6.1}
$$
The estimate for the velocity vector follows from the Poincaré inequality ([36], Part 1, § 116) for the function $\widetilde{\boldsymbol{v}}(\boldsymbol{k}\varepsilon+ \varepsilon\boldsymbol{y},t)=\boldsymbol{u}^{\boldsymbol{k}}(\boldsymbol{y},t)= \overline{\boldsymbol{u}}^{\,\boldsymbol{k}}(\boldsymbol{z},t)$, which vanishes on the boundary of the domain $\Omega_f^{\boldsymbol{k},\varepsilon}(r)$ in the cube $\varepsilon Y$ with edge length $\varepsilon$:
$$
\begin{equation*}
\begin{aligned} \, &\text{if }\int_{Y}|\boldsymbol{u}^{\boldsymbol{k}}|^2\, dy\leqslant M^2\int_{Y}|\mathbb{D}(y,\boldsymbol{u}^{\boldsymbol{k}})|^2\, dy, \\ &\qquad\text{then }\int_{\varepsilon Y} |\overline{\boldsymbol{u}}^{\,\boldsymbol{k}}|^2\, dy\leqslant \varepsilon^2M^2\int_{\varepsilon Y} |\mathbb{D}(z,\overline{\boldsymbol{u}}^{\,\boldsymbol{k}})|^2\, dy, \end{aligned}
\end{equation*}
\notag
$$
where the constant $M$ is independent of $\varepsilon$.
Thus,
$$
\begin{equation*}
\begin{aligned} \, \int_{\Omega_f^{\boldsymbol{k},\varepsilon}(r)} |\widetilde{\boldsymbol{v}}|^2\,dy &\leqslant \varepsilon^2M^2\int_{\Omega_f^{\boldsymbol{k},\varepsilon}(r)} |\mathbb{D}(y,\widetilde{\boldsymbol{v}})^2|\, dy \\ &=M^2\int_{\Omega_f^{\boldsymbol{k},\varepsilon}(r)} \alpha_{\mu} |\mathbb{D}(y,\widetilde{\boldsymbol{v}})^2|\, dy \leqslant M^2(M_0). \end{aligned}
\end{equation*}
\notag
$$
This yields the estimate for the velocity vector.
To prove the boundedness of the pressure $\widetilde{p}(\boldsymbol{x},t)$ in the space $\mathbb{L}_2(\Omega_T)$, we represent the identity (4.2) in the form
$$
\begin{equation}
\begin{aligned} \, l(\boldsymbol{\varphi}) &\equiv\iint_{\Omega_T}\widetilde{p}\, \nabla\cdot\boldsymbol{\varphi}\,dx\, dt \nonumber \\ &=\iint_{\Omega_T}\nabla\cdot\sqrt{\alpha_{\mu}} \bigl((\mathbb{F}- \mathring{\mathbb{F}})- p^0\,\mathbb{I}\bigr)\cdot \boldsymbol{\varphi}\,dx\, dt \equiv \langle\boldsymbol{f},\boldsymbol{\varphi}\rangle, \end{aligned}
\end{equation}
\tag{6.2}
$$
where
$$
\begin{equation*}
\begin{gathered} \, \boldsymbol{f}=\nabla\cdot\sqrt{\alpha_{\mu}}\bigl((\mathbb{F}- \mathring{\mathbb{F}})- p^0\,\mathbb{I}\bigr),\qquad \mathbb{F}=\chi^{\varepsilon}\sqrt{\alpha_{\mu}}\, \mathbb{D}(x,\widetilde{\boldsymbol{v}}),\quad \mathbb{F},\mathring{\mathbb{F}\,}\in \mathbb{L}_2(\Omega_T), \\ \boldsymbol{f}\in \mathbb{H}^{-1},\qquad \mathbb{H}= \mathring{\mathbb{W}}^1_2(\Omega_T),\qquad \mathbb{H}^{-1}=\mathbb{L}_2\bigl((0,T); \mathring{\mathbb{W}}^{-1}_2(\Omega)\bigr) \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation}
\begin{aligned} \, \|\boldsymbol{f}\|_{\mathbb{H}^{-1}} &= \sup_{\boldsymbol{\varphi}\in\mathbb{H}} \frac{\bigl|\iint_{\Omega_T}\sqrt{\alpha_{\mu}}\, ((\mathbb{F}-\mathring{\mathbb{F}\,})-p^0\,\mathbb{I}): \mathbb{D}(x,\boldsymbol{\varphi})\,dx\,dt\bigr|} {\|\mathbb{D}(x,\boldsymbol{\varphi})\|_{2,\Omega_T}} \nonumber \\ &\leqslant\bigl\|\sqrt{\alpha_{\mu}}\bigl((\mathbb{F}-\mathring{\mathbb{F}}) - p^0\,\mathbb{I}\bigr)\bigr\|_{2,\Omega_T} \leqslant M_{p}. \end{aligned}
\end{equation}
\tag{6.3}
$$
The equality (6.2) and the estimate (6.3) mean that the linear functional is bounded (see [20], Ch. IV, § 1) and its norm does not exceed $M_{p}$. This is equivalent to the estimate (5.2) for the pressure. $\Box$ 6.2. A priori bounds for weak solutions of the diffusion-convection problem $\mathbb{B}^{\varepsilon}(r)$Lemma 6.2. The diffusion-convection problem $\mathbb{B}^{\varepsilon}(r)$ has a unique weak solution satisfying the estimates (5.3)–(5.5). Proof. We use the Leray–Schauder fixed point theorem (see Theorem 2.3).
Put $\mathfrak{N}=\{u\in \mathbb{L}_2(\Omega_{f,T}(r))\colon \|c\|_{2,\Omega_{f,T}(r)}\leqslant M(M_0)\}$. We easily see that this set is closed in the metric of $\mathbb{V}_2(\Omega_{f,T}(r))$. Define an operator $s^{\tau}\,{=}\,\boldsymbol{\Psi}^{\tau}(u)$ sending every function $u\,{\in}\,\mathfrak{N}$ to the solution $s^{\tau}$ of the modified diffusion-convection problem $\mathbb{B}(r)$ with the function $\psi(c)$ replaced by $\tau\psi(u)$. This linear problem is referred to as Problem $\mathbb{B}^{\tau}(r,u)$. By construction, the solution $s^{\tau}$ of Problem $\mathbb{B}^{\tau}(r,u)$ satisfies an integral identity
$$
\begin{equation}
\begin{aligned} \, &\int_{\Omega_f(t_0)}s^{\tau}(\boldsymbol{x},t_0)\xi(\boldsymbol{x},t_0)\, dx- \int_{\Omega(0)} s^{\tau}(\boldsymbol{x},0)\xi(\boldsymbol{x},0)\, dx \nonumber \\ &\qquad+\int_0^{t_0}\int_{\Omega_f(t)}\biggl(-s^{\tau}\, \frac{\partial \xi}{\partial t} +\nabla\xi\cdot\bigl(d_0\,\nabla\,s^{\tau}- \tau\widetilde{\boldsymbol{v}}\psi(u)\bigr) \biggr) \,dx\,dt=0, \end{aligned}
\end{equation}
\tag{6.5}
$$
which holds for all $t_0$, $0\leqslant t_0\leqslant T$, and for all smooth functions $\xi$ vanishing on the boundary $S^1\cup S^2$ for $t\geqslant 0$.
The linear problem $\mathbb{B}^{\tau}(r,u)$ has a unique weak solution $s^{\tau}$ in $\mathbb{V}_2(\Omega_{f,T}(r))$ and the estimates (5.3) and (5.4) hold for its solutions. This assertion is simple, but we were unable to find a proof in the literature although the solubility can easily be proved using the Galerkin method and the methods developed in [25]. It follows from the estimates (5.3) and (5.4) for $s^{\tau}$ and Theorem 2.2 that the operator $\boldsymbol{\Psi}^{\tau}(u)$ is completely continuous and maps the set $\mathfrak{N}$ to itself. When $\tau=0$, Problem $\mathbb{B}^0(r,u)$ has a solution $s^0$. Therefore, the operator $\boldsymbol{\Psi}^{\tau}(u)$ has at least one fixed point $s^1=c$, which is a weak solution of the modified diffusion-convection problem $\mathbb{B}^1(r,c)$. We are now in a position to prove the estimate (5.5) for the solutions $c$ of the modified diffusion-convection problem $\mathbb{B}(r)$. We put $c=u+c^*$ and $\xi=u^+=\max\{u,0\}$ in the identity (4.6). By definition, $u^+\geqslant 0$, $u^+u=|u^+|^2$, $\nabla u\cdot\nabla u^+=|\nabla u^+|^2$, $u^+=0$ on $S^1\cup S^2$ and $(\partial u^+/\partial t)u=(\partial u/\partial t)u^+$. Taking (4.5) into account, we arrive at the following chain of equalities:
$$
\begin{equation*}
\begin{gathered} \, \begin{aligned} \, &\int_{\Omega_f(t_0)}\bigl(c^*+u(\boldsymbol{x},t_0)\bigr) u^+(\boldsymbol{x},t_0)\, dx=\int_{\Omega_f(t_0)}\bigl(c^*u^+(\boldsymbol{x},t_0)+ |u(\boldsymbol{x},t_0)|^2\bigr)\,dx \\ &\qquad=\int_0^{t_0}\int_{\Omega_f(t)}(u+c^*)\, \frac{\partial u^+}{\partial t}\, dx\, dt-I_1+I_2 \\ &\qquad=\int_0^{t_0}\int_{\Omega_f(t)}\frac{\partial }{\partial t} \biggl(c^*u^++ \frac{1}{2}\,|u^+|^2\biggr)\, dx\, dt+I_1-I_2 \\ &\qquad=\int_{\Omega_f(t_0)}\biggl(c^*u^+(\boldsymbol{x},t_0)+ \frac{1}{2}\,|u^+(\boldsymbol{x},t_0)|^2\biggr)\, dx +I_{\Gamma}+I_1-I_2, \end{aligned} \\ \begin{aligned} \, I_1 &=d_0\int_0^{t_0}\int_{\Omega_f(t)}\nabla u^+\cdot\nabla u\,dx\, dt= d_0\int_0^{t_0}\int_{\Omega_f(t)}|\nabla u^+|^2\,dx\,dt, \\ I_2 &=\int_0^{t_0}\int_{\Omega_f(t)}\nabla u^+\cdot \widetilde{\boldsymbol{v}}\psi(u^++c^*)\, dx\, dt, \\ I_{\Gamma} &=\int_0^{t_0}\int_{\Gamma(t)}\biggl(c^*u^++ \frac{1}{2}|u^+|^2\biggr) D^{\varepsilon}_{N}\sin\varphi\,d \sigma\, dt. \end{aligned} \end{gathered}
\end{equation*}
\notag
$$
Since $D^{\varepsilon}_{N}\sin\varphi\geqslant 0$, we finally obtain the inequality
$$
\begin{equation}
\frac{1}{2}\int_{\Omega_f(t_0)}|u^+(\boldsymbol{x},t_0)|^2\, dx \leqslant -\int_0^{t_0}\int_{\Omega_f(t)}\nabla u^+\cdot \widetilde{\boldsymbol{v}}\psi(u+c^*)\, dx\, dt,
\end{equation}
\tag{6.6}
$$
which yields that $u^+(\boldsymbol{x},t)=0$ almost everywhere in $\Omega_{f,T}$.
Indeed, otherwise the left-hand side of (6.6) is strictly positive on the set $Q_{u}$ where $u>0$ (or $c>c^*$) while the right-hand side vanishes. Hence, $c\leqslant c^*$. The case when $c\geqslant 0$ can be considered in a similar way. The estimates (5.5) for the solutions of the modified diffusion-convection problem $\mathbb{B}^{\varepsilon}(r)$ show that $\psi(s)=s+\theta/\delta$ in the identity (4.6). Hence this problem coincides with the diffusion-convection problem $\mathbb{B}^{\varepsilon}(r)$, and the extended functions $\widetilde{\boldsymbol{v}}^{\,\varepsilon}$ and $\widetilde{c}^{\,\varepsilon}$ satisfy the identity
$$
\begin{equation}
\begin{aligned} \, &\int_{\Omega}\chi\biggl(r,\frac{\boldsymbol{x}}{\varepsilon}\biggr) \widetilde{c}^{\,\varepsilon}(\boldsymbol{x},t_0) \chi\biggl(r(\boldsymbol{x},t_0), \frac{\boldsymbol{x}}{\varepsilon}\biggr)\, dx- \int_{\Omega}\chi\biggl(r(\boldsymbol{x},0), \frac{\boldsymbol{x}}{\varepsilon}\biggr) c^0(\boldsymbol{x})\xi(\boldsymbol{x},0)\, dx \nonumber \\ &\qquad+\int_0^{t_0}\int_{\Omega}\chi \biggl(r,\frac{\boldsymbol{x}}{\varepsilon}\biggr) \biggl(-\widetilde{c}^{\,\varepsilon}\, \frac{\partial \xi}{\partial\tau}+ \nabla\xi\cdot\biggl(d_0\nabla\widetilde{c}^{\,\varepsilon}- \widetilde{\boldsymbol{v}}^{\,\varepsilon} \biggl(\widetilde{c}^{\,\varepsilon} + \frac{\theta}{\delta}\biggr)\biggr)\biggr)\, dx\, d\tau=0, \end{aligned}
\end{equation}
\tag{6.7}
$$
which holds for all smooth functions $\xi$ vanishing on the boundary $S^1\cup S^2$ of the domain $\Omega$ for $t>0$. $\Box$ Remark 6.1. In particular, (6.7) means that $\partial\widetilde{c}^{\,\varepsilon}/\partial t\in \mathbb{L}_2\bigl((0,T);\mathbb{W}^{-1}_2(\Omega)\bigr)$ and § 7. Proof of Theorem 5.27.1. Homogenization: the choice of convergent subsequencesThe estimates (5.2)–(5.4) enable us to choose subsequences converging as $\varepsilon\to 0$ (we preserve the same notation for simplicity):
$$
\begin{equation}
\begin{gathered} \, \widetilde{\boldsymbol{v}}^{\,\varepsilon}\rightharpoonup\boldsymbol{v},\qquad \widetilde{\boldsymbol{v}}^{\,\varepsilon}\xrightarrow{\textrm{t.-s.}} \boldsymbol{V},\qquad (1-\chi^{\varepsilon})\widetilde{\boldsymbol{v}}^{\,\varepsilon}\quad \text{converges strongly in }\mathbb{L}_2(\Omega_T)\text{ to zero}; \nonumber \\ \varepsilon\chi^{\varepsilon} \mathbb{D}(x,\widetilde{\boldsymbol{v}}^{\,\varepsilon}) \xrightarrow{\textrm{t.-s.}}\mathbb{D}(y,\boldsymbol{V}),\qquad \overline{p}^{\,\varepsilon}\rightharpoonup p,\qquad \widetilde{p}^{\,\varepsilon}\xrightarrow{\textrm{t.-s.}}P; \nonumber \\ \sqrt{\alpha_{\mu}}\,\mathring{\mathbb{F}}^{\varepsilon}\quad \text{converges strongly in } \mathbb{L}_2(\Omega_T)\text{ to zero}; \nonumber \\ \begin{gathered} \, \mathring{\boldsymbol{v}}^{\varepsilon}\quad \text{converges strongly in } \mathbb{L}_2(\Omega_T)\text{ to zero}; \\ \widetilde{c}^{\,\varepsilon}\quad\text{converges strongly in } \mathbb{L}_2(\Omega_T)\text{ to } c, \qquad \chi^{\varepsilon}\widetilde{c}^{\,\varepsilon} \xrightarrow{\textrm{t.-s.}}mc; \\ \nabla\widetilde{c}^{\,\varepsilon}\xrightarrow{\textrm{t.-s.}} \nabla_{x}c+\nabla_{y}C, \end{gathered} \end{gathered}
\end{equation}
\tag{7.1}
$$
where the functions $\boldsymbol{V}=\boldsymbol{V}(\boldsymbol{x},t,\boldsymbol{y})$, $\boldsymbol{P}=\boldsymbol{P}(\boldsymbol{x},t,\boldsymbol{y})$ and $\boldsymbol{C}=\boldsymbol{C}(\boldsymbol{x},t,\boldsymbol{y})$ are $1$-periodic in the variable $\boldsymbol{y}$,
$$
\begin{equation*}
\boldsymbol{v},p\in\mathbb{L}_2(\Omega_T),\qquad c\in\mathbb{W}^{1,0}_2(\Omega_T),\qquad P,\boldsymbol{V},C,\mathbb{D}(y,\boldsymbol{V}),\qquad \nabla_{y}\,C\in\mathbb{L}_2(,\Omega_T\times Y)
\end{equation*}
\notag
$$
and
$$
\begin{equation}
\begin{gathered} \, \|\boldsymbol{v}\|_{2,\Omega_T}+\|p\|_{2,\Omega_T}\leqslant M_{p}, \\ \|P\|_{2,Y\times\Omega_T}+\|\boldsymbol{V}\|_{2,Y\times\Omega_T}+ \|\mathbb{D}(y,\boldsymbol{V})\|_{2,Y\times\Omega_T}\leqslant M_{p}, \\ 0\leqslant c(\boldsymbol{x},t)\leqslant c^*,\qquad (\boldsymbol{x},t)\in \Omega_T, \\ \|\nabla c\|_{2,\Omega_T}+\|\nabla C\|_{2,Y\times\Omega_T} \leqslant M_{c}. \end{gathered}
\end{equation}
\tag{7.2}
$$
7.2. Homogenization of the dynamical problem $\mathbb{B}^{\varepsilon}(r)$ Lemma 7.1. Under the hypotheses of Theorem 5.2, we have the macroscopic continuity equation Proof. Letting $\varepsilon\to 0$ in the identity (4.3) with smooth test functions $\psi$ vanishing on the boundary $S^1\cup S^2$ of the domain $\Omega$ for $t>0$, we arrive at an integral identity
$$
\begin{equation}
0=\iint_{\Omega_T}\biggl(-\delta m\,\frac{\partial\psi}{\partial t} +\boldsymbol{v}\cdot\nabla\psi\biggr)\,dx\,dt= \iint_{\Omega_T}\biggl(\delta\,\frac{\partial m}{\partial t}\,\psi+ \boldsymbol{v}\cdot\nabla\psi\biggr)\,dx\,dt.
\end{equation}
\tag{7.4}
$$
The identity (7.4) means that the differential equation (7.3) and the first boundary condition in (4.11) hold in the ordinary sense. $\Box$ Lemma 7.2. Under the hypotheses of Theorem 5.2, the limiting functions $\boldsymbol{v}$ and $p$ are bounded in the spaces $\mathbb{L}_2(\Omega_T)$ and $\mathbb{W}^{1,0}_2(\Omega_T)$ respectively: The symmetric strictly positive definite matrix $\mathbb{C}^v(r)$ is given by (4.14). Proof. We claim that
$$
\begin{equation}
P(\boldsymbol{x},t,\boldsymbol{y})= p(\boldsymbol{x},t) \chi(\boldsymbol{x},t,\boldsymbol{y}).
\end{equation}
\tag{7.7}
$$
Indeed, put $\boldsymbol{\varphi}=\varepsilon\varphi_0(\boldsymbol{x},t) \boldsymbol{\varphi}_1 (\boldsymbol{x}/\varepsilon)$ in the identity (4.2), where $\varphi_0(\boldsymbol{x},t)$ is an arbitrary smooth function on $\Omega_T$ vanishing on the lateral boundary of the domain $\Omega_T$ and $\boldsymbol{\varphi}_1(\boldsymbol{y})$ is an arbitrary smooth function on $Y$. Letting $\varepsilon \to 0$, we have
$$
\begin{equation*}
\iint_{\Omega_T}\varphi_0(\boldsymbol{x},t) \biggl(\int_{Y}P(\boldsymbol{x},t,\boldsymbol{y})\nabla_{y} \cdot\boldsymbol{\varphi}_1(\boldsymbol{y})\, dy\biggr)\, dx\, dt=0,
\end{equation*}
\notag
$$
which is equivalent to (7.7).
We easily can derive the continuity equation and the boundary condition for the velocity vector $\boldsymbol{V}$:
$$
\begin{equation}
\nabla_{y}\cdot\boldsymbol{V}=0,\qquad \boldsymbol{y}\in Y_f(r),\quad \boldsymbol{V}=0, \quad |\boldsymbol{y}|=r,\quad 0<t<T.
\end{equation}
\tag{7.8}
$$
To derive the continuity equation, take $\psi=\varepsilon\psi_0(\boldsymbol{x},t) \psi_1(\boldsymbol{x}/\varepsilon)$ for the test function in (4.3) and let $\varepsilon\to 0$.
To derive the boundary condition in (7.8), we pass to a two-scale limit in the equality $(1-\chi^{\varepsilon})\widetilde{\boldsymbol{v}}^{\,\varepsilon}=0$. This yields the relation $(1- \chi(\boldsymbol{x},t,\boldsymbol{y})) \boldsymbol{V} (\boldsymbol{x},t,\boldsymbol{y})=0$. Then, using the smoothness $\boldsymbol{V}\in \mathbb{W}^{1,0}_2(Y)$ of the function $\boldsymbol{V}$, we arrive at the boundary condition in (7.8). Letting $\varepsilon\rightarrow 0$ in (4.2), we obtain the homogenized dynamical equation
$$
\begin{equation}
\begin{aligned} \, &\int_0^{T}\int_{\Omega}\biggl(\int_{Y_f(r)} \bigl(\mu_1 \mathbb{D}(y,\boldsymbol{V}): \mathbb{D}(y,\boldsymbol{\varphi})-p\bigr) \nabla\cdot\boldsymbol{\varphi}\,dy\biggr)\, dx\, dt \nonumber \\ &\qquad=-\int_0^{T}\int_{\Omega}\nabla p^0\cdot\boldsymbol{\varphi}\,dx\, dt. \end{aligned}
\end{equation}
\tag{7.9}
$$
Put $\boldsymbol{\varphi}=\varphi_0(\boldsymbol{x},t) \boldsymbol{\varphi}_i(\boldsymbol{x}/\varepsilon)$ in (7.9), where the solenoidal functions $\boldsymbol{\varphi}_i(\boldsymbol{y})$ vanish on the boundary $\gamma(r)$ of the liquid component $Y_f(r)$ of the domain $Y$, $\boldsymbol{\varphi}_i\in \mathbb{W}^1_2(Y_f(r))$, $\operatorname{supp}\boldsymbol{\varphi}_i\in Y_f(r)$, $\varphi_i(\boldsymbol{x},t)=0$ on the boundary $S^0$ of the domain $\Omega$ for $t>0$, $\operatorname{supp}\boldsymbol{\varphi}_i\subset Y_f(r)$ and $\langle\boldsymbol{\varphi}_i\rangle_{Y_f}=\boldsymbol{e}_i$, where $\{\boldsymbol{e}_1,\boldsymbol{e}_2,\boldsymbol{e}_3\}$ is an orthonormal basis in $\mathbb{R}^3$. This choice is possible by Lemma 2.1.
We have
$$
\begin{equation}
\iint_{\Omega_T}\biggl(\varphi_0 a_i-p\, \frac{\partial\varphi_0}{\partial x_i}\biggr)\, dx\, dt= -\int_0^{T}\int_{\Omega}\nabla p^0\,dx\, dt,
\end{equation}
\tag{7.10}
$$
where $a_i=\int_{Y_f}\mu_1\mathbb{D}(y,\boldsymbol{V}): \mathbb{D}(y,\boldsymbol{\varphi}_{1,i})\,dy \in \mathbb{L}_2(\Omega_T)$.
Therefore, $ a_i=-\partial p/\partial x_i$ and
$$
\begin{equation*}
\|\nabla p\|_{2,\Omega_T}\leqslant M\sum_{i=1}^3 \biggl\|\int_{Y_f} \mu_1\mathbb{D}(y,\boldsymbol{V}): \mathbb{D}(y,\boldsymbol{\varphi}_{1,i})\, dy\biggr\|_{2,\Omega_T} \leqslant M_{p}.
\end{equation*}
\notag
$$
This proves the estimate (7.5) for the pressure.
Since the functions $\varphi_0$ in the identity (4.2) are equal to zero only on the part $S^0$ of the boundary $S$ for $t>0$, we see that $p$ must vanish on the part $S^1\cup S^2$ of the boundary $S$ for $t>0$. This proves the second boundary condition in for the pressure in (4.11). Since the pressure belongs to the space $\mathbb{W}^{1,0}_2(\Omega_T)$, the relations (7.8) can be rewritten in the form of the Stokes equations
$$
\begin{equation*}
\begin{gathered} \, \mu_1\Delta_{y}\boldsymbol{V}-\nabla_{y}\Pi= \nabla(p+p^0),\quad \nabla_{y}\cdot\boldsymbol{V}=0,\qquad \boldsymbol{y}\in Y_f, \\ \boldsymbol{V}(\boldsymbol{x},t,\boldsymbol{y})=0,\qquad \boldsymbol{y}\in\gamma(r). \end{gathered}
\end{equation*}
\notag
$$
The solution of this system is given by an explicit formula
$$
\begin{equation*}
\boldsymbol{V}=-\frac{1}{\mu_1}\sum_{i=1}^3(\boldsymbol{V}^i \otimes\boldsymbol{e}_i)\cdot\nabla(p+p^0),
\end{equation*}
\notag
$$
where the functions $\boldsymbol{V}^i(\boldsymbol{y})$ are defined by (4.14).
Thus,
$$
\begin{equation*}
\boldsymbol{v}=\langle\boldsymbol{V}\rangle_{Y_f}= -\frac{1}{\mu_1} \biggl\langle\sum_{i=1}^3(\boldsymbol{V}^i\otimes \boldsymbol{e}_i)\biggr\rangle_{Y_f}\cdot \nabla(p+p^0) =-\frac{1}{\mu_1}\,\mathbb{C}^v\cdot\nabla(p+p^0).
\end{equation*}
\notag
$$
This completes the proof of Lemma 7.2. $\Box$ Corollary 7.1. The limiting functions $\boldsymbol{v}(\boldsymbol{x},t)$ and $p(\boldsymbol{x},t)$ are uniquely determined. Proof. This follows from the estimate (7.2) for the difference of two solutions of Darcy’s system of differential equations (7.3), (7.6) completed with the boundary conditions (4.11). $\Box$ 7.3. Homogenization of the diffusion-convection problem $\mathbb{B}^{\varepsilon}(r)$ Lemma 7.3. Under the hypotheses of Theorem 5.2, the diffusion-convection problem $\mathbb{H}(r)$ has a unique weak solution $c\in \mathbb{V}_2(\Omega_T)$ such that The symmetric strictly positive definite matrix $\mathbb{C}^c(r)$ is defined by (4.14). To prove the identity (7.12), let $\varepsilon \to 0$ in the identity (4.4). The estimates (7.11) follow from (5.3) and (5.5). The properties of the matrices $\mathbb{C}^v$ and $\mathbb{C}^c$ were studied in [15] in the case when the structure of the pore space is fixed. In our situation, these matrices will only be positive definite for all $r(\boldsymbol{x},t)>0$. Hence we need to separately prove a uniform lower bound for the eigenvalues of the matrices $\mathbb{C}^v$ and $\mathbb{C}^c$ for all $r(\boldsymbol{x},t)>0$. Lemma 7.4. Under the hypotheses of Theorem 5.2, the symmetric strictly positive definite matrices $\mathbb{C}^v(r)$ and $\mathbb{C}^c(r)$ are infinitely smooth with respect to the parameter $r$ and we have with a strictly positive constant $\nu_0$ for all $\boldsymbol{\zeta}\in \mathbb{R}^3$, $|\boldsymbol{\zeta}|=1$. Proof. First of all, we consider the matrix $\mathbb{C}^v(r)$, prove the solubility of the problem (4.14) for $r(\boldsymbol{x},t)=0$ and study the properties of its solutions $\boldsymbol{V}^i(r;\boldsymbol{y})$ for all $r(\boldsymbol{x},t)\geqslant 0$.
Recall that the matrix $\mathbb{C}^v(r)$ is defined in terms of the solutions $\boldsymbol{V}^i(r;\boldsymbol{y})$ of the boundary-value problem (4.14):
$$
\begin{equation}
\begin{gathered} \, \Delta_{y}\boldsymbol{V}^i-\nabla_{y}\Pi^i=-\boldsymbol{e}^i,\quad \nabla_{y}\cdot\boldsymbol{V}^i=0,\qquad |\boldsymbol{y}|>r, \\ \boldsymbol{V}^i(\boldsymbol{y})=0,\qquad |\boldsymbol{y}|=r,\qquad \int_{Y_f}\Pi^i(\boldsymbol{y})\, dy=0 \end{gathered}
\end{equation}
\tag{7.14}
$$
by the formula
$$
\begin{equation*}
\mathbb{C}^v(r)=2\sum_{i=1}^3\langle\boldsymbol{V}^i\rangle_{Y_f} \otimes\boldsymbol{e}^i.
\end{equation*}
\notag
$$
We easily see that the entries of the matrix $\mathbb{C}^v(r)$ are infinitely smooth functions of the parameter $r$.
Multiplying the dynamical Stokes equation in (7.14) by $\boldsymbol{V}^j$ and integrating by parts over $Y_f(r)$, we have
$$
\begin{equation}
\langle\mathbb{D}(y,\boldsymbol{V}^j): \mathbb{D}(y,\boldsymbol{V}^i)\rangle_{Y_f}= \langle\boldsymbol{V}^j\cdot\boldsymbol{e}^i\rangle_{Y_f},\qquad i,j=1,2,3.
\end{equation}
\tag{7.15}
$$
It follows from (7.15) that the matrix $\mathbb{C}^v(r)$ is symmetric.
When $i=j$, these equalities and the Poincaré–Friedrichs inequality ([36], Vol. 4, Ch. III, § 116) yield that, for an arbitrary positive $\nu$,
$$
\begin{equation*}
\begin{aligned} \, &\int_{Y_f(r)}|\mathbb{D}(y,\boldsymbol{V}^i)|^2\, dy= \biggl|\biggl(\int_{Y_f(r)}\boldsymbol{V}^i_{r}\, dy\biggr)\cdot \boldsymbol{e}^i\biggr| \\ &\qquad\leqslant\nu\int_{Y_f(r)}|\boldsymbol{V}^i|^2\, dy+ \frac{1}{4\nu}\,m_{r}\leqslant \nu M_3\int_{Y_f(r)} |\mathbb{D}(y,\boldsymbol{V}^i)|^2\, dy+ \frac{1}{4\nu}\,m(r), \end{aligned}
\end{equation*}
\notag
$$
where the constant $M_3$ in the Poincaré–Friedrichs inequality is independent of $r$.
Choosing $\nu M_3=1/2$, we have
$$
\begin{equation}
\int_{Y_f(r)}\bigl(|\boldsymbol{V}^i|^2+ |\mathbb{D}(y,\boldsymbol{V}^i)|^2\bigr)\, dy \leqslant M_3 m(r).
\end{equation}
\tag{7.16}
$$
The functions $\boldsymbol{V}^i$ and $\mathbb{D}(y,\boldsymbol{V}^i)$ are continuous with respect to the parameter $r$ and
$$
\begin{equation*}
\lim_{r\to 0}\boldsymbol{V}^i=\boldsymbol{V}^i_0,\qquad \lim_{r\to 0}\mathbb{D}(y,\boldsymbol{V}^i)=\mathbb{D}(y,\boldsymbol{V}^i_0),
\end{equation*}
\notag
$$
where
$$
\begin{equation}
\begin{gathered} \, \Delta_{y}\boldsymbol{V}_0^i-\nabla_{y}\Pi_0^i=-\boldsymbol{e}_i,\quad \nabla_{y}\cdot\boldsymbol{V}_0^i=0,\qquad |\boldsymbol{y}|>0, \\ \boldsymbol{V}_0^i(r;\boldsymbol{0})=0,\qquad \int_{Y}\Pi^i_0(r;\boldsymbol{y})\, dy=0. \end{gathered}
\end{equation}
\tag{7.17}
$$
We now consider arbitrary constant vectors $\boldsymbol{\zeta}=(\zeta_1,\zeta_2,\zeta_3)$ and $\boldsymbol{\eta}=(\eta_1,\eta_2,\eta_3)$ of unit norm in $\mathbb{R}^3$ and put
$$
\begin{equation*}
\begin{gathered} \, \boldsymbol{z}_{\zeta}(\boldsymbol{y})= \sum_{i=1}^3\zeta_i\boldsymbol{V}^i(\boldsymbol{y}),\qquad \boldsymbol{z}_{\eta}(\boldsymbol{y})= \sum_{j=1}^3\eta_j\boldsymbol{V}^j(\boldsymbol{y}), \\ f(\boldsymbol{\eta})=\bigl(\mathbb{C}^v(r)\cdot\boldsymbol{\eta} \bigr) \cdot\boldsymbol{\eta}= \int_{Y_f(r)}|\mathbb{D}(y,\boldsymbol{z}_{\eta})|^2\, dy. \end{gathered}
\end{equation*}
\notag
$$
The continuous function $f(\boldsymbol{\eta})$ attains its minimum $f^0{=}f(\boldsymbol{\eta^0})$ on the sphere $|\boldsymbol{\eta}|=1$.
Suppose that $f^0=0$. Then $\mathbb{D}(y,\boldsymbol{z}_{\eta^0})=0$. This is possible only when $\boldsymbol{z}_{\eta^0}$ is a linear function of $\boldsymbol{y}$. On the other hand, it follows from the boundary-value problem (7.17) that
$$
\begin{equation*}
\begin{gathered} \, \Delta_{y}\boldsymbol{z}_{\eta^0}-\nabla_{y}\,\Pi_{\eta^0}= -\boldsymbol{\eta^0},\quad \nabla_{y}\cdot\boldsymbol{z}_{\eta^0}=0,\qquad |\boldsymbol{y}|>0, \\ \boldsymbol{z}_{\eta^0}(\boldsymbol{0})=0,\qquad \int_{Y}\Pi_{\eta^0}(\boldsymbol{y})\, dy=0. \end{gathered}
\end{equation*}
\notag
$$
Therefore $\Pi_{\eta^0}=\boldsymbol{\eta}^0\cdot\boldsymbol{y}$ everywhere in the cube $Y$. Since the function $\Pi_{\eta^0}$ is periodic in $\boldsymbol{y}$, this is possible only if $\boldsymbol{\eta^0}=0$. The resulting contradiction proves our claim.
We now derive the formulae (4.14) for the functions $C^i(\boldsymbol{y})$. Consider the limiting identity resulting from (4.4) as $\varepsilon \to 0$ with test functions $\xi=\xi(\boldsymbol{x},t)$:
$$
\begin{equation}
\begin{aligned} \, 0 &=\int_0^{t_0}\int_{\Omega}\biggl(-mc\,\frac{\partial\xi}{\partial t}+ d_0 \nabla\xi\cdot\biggl(\nabla c+\int_{Y_f(r)}\nabla_{y}C \,dy- \boldsymbol{v}\biggl(c+\frac{\theta}{\delta}\biggr)\biggr)\biggr)\, dx\, dt \nonumber \\ &=\iint_{\Omega_T}\biggl(-mc\,\frac{\partial\xi}{\partial t}+d_0\nabla\xi \cdot\biggl(\nabla c +\int_{Y_f(r)}\nabla_{y}C\,dy + \boldsymbol{f}\biggr)\biggr)\, dx\, dt \nonumber \\ &=\int_0^{t_0}\int_{\Omega}\biggl(-mc\,\frac{\partial\xi}{\partial t}+ d_0 \nabla\xi\cdot\bigl(\mathbb{C}^c(r)\cdot\nabla c+ \boldsymbol{f}\bigr)\biggr)\, dx\, dt. \end{aligned}
\end{equation}
\tag{7.18}
$$
Here
$$
\begin{equation*}
\begin{gathered} \, d_0\boldsymbol{f}(\boldsymbol{x},t)=- \biggl(\boldsymbol{v}+ \frac{\theta}{\delta}\biggr) =(f_1,f_2,f_3), \\ \mathbb{C}^c(r)=\mathbb{I}+\int_{Y_f(r)}\nabla_{y}C\,dy= \mathbb{I}+\mathbb{C}_0^c(r). \end{gathered}
\end{equation*}
\notag
$$
Then we choose $\xi=\varepsilon\xi_0(\boldsymbol{x},t) \xi_1(\boldsymbol{x}/\varepsilon)$ for test functions.
Letting $\varepsilon \to 0$, we obtain an identity
$$
\begin{equation*}
\int_0^{t_0}\int_{\Omega}\xi_0\biggl(\int_{Y_f}\nabla_{y}\xi_1\cdot (\nabla_{y} C+\boldsymbol{f})dy\biggr)\, dx\, dt=0.
\end{equation*}
\notag
$$
It follows that
$$
\begin{equation}
\nabla_{y}\cdot(\nabla_{y}C+\boldsymbol{f})=0,\quad \boldsymbol{y}\in Y_f,\qquad \frac{\partial C}{\partial n}+ \boldsymbol{f}\cdot\boldsymbol{n}=0,\quad |\boldsymbol{y}|=r,
\end{equation}
\tag{7.19}
$$
where $\boldsymbol{n}=\boldsymbol{y}/r$.
As usual, we use the representation
$$
\begin{equation*}
C(r;\boldsymbol{x},t,\boldsymbol{y})= \sum_{i=1}^3C^i(r;\boldsymbol{y}) f_i(\boldsymbol{x},t).
\end{equation*}
\notag
$$
We have
$$
\begin{equation}
\nabla_{y}\,\cdot(\nabla_{y}C^i+\boldsymbol{e}^i)=0,\quad \boldsymbol{y}\in Y_f,\qquad \frac{\partial C^i}{\partial n}+ \boldsymbol{e}^i\cdot\boldsymbol{n}=0, \quad |\boldsymbol{y}|=r.
\end{equation}
\tag{7.20}
$$
Multiplying the differential equation for $C^i$ in (7.20) by $C^j$ and integrating by parts over the domain $Y_f(r)$, we arrive at the equalities
$$
\begin{equation}
\int_{Y_f(r)}\nabla_{y}C^i\cdot\nabla_{y}\,C^j\,dy+ \int_{Y_f(r)}\nabla_{y}C^j\cdot\boldsymbol{e}^i\,dy=0,\qquad i,j,=1,2,3.
\end{equation}
\tag{7.21}
$$
In particular,
$$
\begin{equation*}
\int_{Y_f(r)}|\nabla_{y}C^i|^2\,dy+ \int_{Y_f(r)}\nabla_{y} C^i\cdot\boldsymbol{e}^i\,dy=0,\qquad i,=1,2,3.
\end{equation*}
\notag
$$
This yields an a priori estimate
$$
\begin{equation*}
\begin{gathered} \, \sum_{i=1}^3\int_{Y_f(r)} \! |\nabla_{y}C^i|^2\,dy= \biggl|\sum_{i=1}^3\int_{Y_f(r)}\nabla_{y}C^i\cdot \boldsymbol{e}^i\,dy\biggr|\leqslant \frac{1}{2}\sum_{i=1}^3\int_{Y_f(r)}|\nabla_{y}C^i|^2\,dy+\frac{3}{2}m(r), \\ \sum_{i=1}^3\int_{Y_f(r)}|\nabla_{y}C^i|^2\,dy \leqslant M_4m(r), \end{gathered}
\end{equation*}
\notag
$$
where the constant $M_4$ is independent of the choice of the function $r\in \mathfrak{M}_T$. Hence the boundary-value problem (7.20) is uniquely soluble in the space $\mathbb{W}^1_2(Y_f(r))$ for all $r>0$.
Moreover, it follows from (7.20) that
$$
\begin{equation}
\bigl(\mathbb{C}_0^c(r)\cdot\boldsymbol{e}^i\bigr)\cdot\boldsymbol{e}^j+ \int_{Y_f(r)}\nabla_{y}C^i\cdot\nabla_{y}\,C^j\,dy=0,\qquad i,j,=1,2,3.
\end{equation}
\tag{7.22}
$$
Hence the matrix $\mathbb{C}_0^c(r)$ (and, therefore, $\mathbb{C}^c(r)$) is symmetric.
The proof of strict positive definiteness for the matrix $\mathbb{C}^c(r)$ is completely analogous to that for $\mathbb{C}^v(r)$. This completes the proof of Lemma 7.4. $\Box$ Corollary 7.2. We can always assume that the matrices $\mathbb{C}^v(r)$ and $\mathbb{C}^c(r)$ are diagonal. To prove this, it suffices to use Theorem 5.28 in [38], Ch. 5, § 6, saying that two symmetric matrices one of which is positive definite are simultaneously orthogonally diagonalizable. In this case, the positive definite matrix is reduced to the identity matrix. We assume throughout that the Cartesian coordinate system $\{\boldsymbol{e}_1, \boldsymbol{e}_2,\boldsymbol{e}_3\}$ is an orthonormal system of eigenvectors of the matrices $\mathbb{C}^v(r)= \mathbb{I}$ and $\mathbb{C}^c(r)=\sum_{i=1}^3c^i(r)\boldsymbol{e}_i\otimes\boldsymbol{e}_i$. Moreover, since the set $Y_f$ is symmetric with respect to orthogonal transformations, we have
$$
\begin{equation*}
c^1(r)=c^2(r)=c^3(r)=\frac{1}{d_0}s(r)\geqslant s_0=\mathrm{const}>0.
\end{equation*}
\notag
$$
Corollary 7.3. The solution $\{\boldsymbol{v},p,c\}$ of Problem $\mathbb{H}(r)$ is a solution of the following initial-boundary value problem: The initial-boundary value problem for the limiting concentration $c$ is understood in the sense of distributions as the integral identity (4.12) along with the boundary condition (7.24) on the boundary $S_T^1$. 7.4. Differential properties of solutions of Problem $\mathbb{H}(r)$Lemma 7.5. Under the hypotheses of Theorem 5.2, the solutions $\boldsymbol{v}$ and $\nabla p$ of the problem (7.23) belong to $\mathbb{H}^{\gamma,\gamma/2}(\overline{\Omega}_T)$ and the following a priori estimate holds: Proof. The function $p(\boldsymbol{x},t)$ satisfies the following boundary-value problem in the domain $\Omega$ for $t>0$:
$$
\begin{equation}
\begin{gathered} \, \Delta p=f,\qquad f=-\mu_1\delta\,\frac{\partial m}{\partial t}, \\ p(\boldsymbol{x},t)=0,\quad\boldsymbol{x}\in S^1\cup S^2,\quad t>0,\qquad \frac{\partial p}{\partial N}(\boldsymbol{x},t)=0,\quad \boldsymbol{x}\in S^0,\quad t>0; \\ f(\,{\cdot}\,,t)\in \mathbb{H}^{(1+\gamma)/2}[0,T]. \end{gathered}
\end{equation}
\tag{7.26}
$$
Let $\widetilde{p}(\boldsymbol{x},t)$ be the extension of $p(\boldsymbol{x},t)$ across the boundary $S=\partial\Omega$ to the domain $Q=\{\boldsymbol{x}\in\mathbb{R}^3\colon |x_i|<1,\,i=1,2,3\}$ in an odd or even way in the direction of the normal vector to $S$. Then $\widetilde{p}$ coincides with a shift of $\pm p$ on every unit cube in $Q\setminus \Omega$. At the same time, the extension $\widetilde{p}(\boldsymbol{x},t)$ satisfies the Poisson equation in the domain $Q$ for $t>0$: On the boundary $S_0^0=\{x_2=\pm 1\}\cup\{x_3=\pm 1\}$ of the domain $Q$ we have and on the boundary $S_0^i=\{x_1=(\pm 1)^i\}$ of the domain $Q$ we have In view of the boundary condition (7.29) and the inequality
$$
\begin{equation*}
\max_{0<t<T}\|\widetilde{f}(\,{\cdot}\,,t)\|_{q,Q}\leqslant M(M_{p}),
\end{equation*}
\notag
$$
the solutions of the boundary-value problem (7.27)–(7.29) satisfy an inner estimate ([37], Ch. 9, § 11, Lemma 9.12) where $q>1$ is arbitrary.
Put $\gamma=(q-3)/3$. Then
$$
\begin{equation}
|\nabla\widetilde{p}\,|^{(0)}_{\Omega_T}+ \langle\nabla\widetilde{p}\,\rangle_{x,\Omega_T}^{(\gamma)} \leqslant \max_{0<t<T} \|\nabla\widetilde{p}(\,{\cdot}\,,t)\|^{(1)}_{q,\Omega}\leqslant M(\gamma,M_{p})
\end{equation}
\tag{7.31}
$$
([25], Ch. II, § 2, Theorem 2.1). This proves that the pressure gradient is bounded and gives its Hölder norm with respect to the spatial variables.
To estimate the Hölder norm of the pressure gradient with respect to time, we consider the finite differences
$$
\begin{equation*}
\begin{aligned} \, D_{h}\widetilde{p}(\boldsymbol{x},t) &=\frac{1}{h^{\gamma/2}} \bigl(\widetilde{p}(\boldsymbol{x},t+h)-\widetilde{p}(\boldsymbol{x},t)\bigr), \\ D_{h}\widetilde{f}(\boldsymbol{x},t) &=\frac{1}{h^{\gamma/2}} \bigl(\widetilde{f}(\boldsymbol{x},t+h)-\widetilde{f}(\boldsymbol{x},t)\bigr). \end{aligned}
\end{equation*}
\notag
$$
They satisfy the Poisson equation (7.27) in $Q$ and the boundary conditions (7.28) and (7.29) on the boundary of $Q$.
Therefore, the estimates (7.30) and (7.31) hold for $D_{h}\widetilde{p}$;
$$
\begin{equation*}
|D_{h}\nabla\widetilde{p}(\boldsymbol{x},t)|\leqslant |D_{h}\nabla\widetilde{p}|^{(0)}_{\Omega_T}\leqslant M(q,M_{p}).
\end{equation*}
\notag
$$
Letting $h\to 0$ in the last inequality, we obtain an estimate
$$
\begin{equation*}
\langle\nabla p\rangle^{(\gamma/2)}_{t,\Omega_T}\leqslant M(q,M_{p}).
\end{equation*}
\notag
$$
Combining it with (7.31) and Darcy’s law (7.23), we complete the proof. $\Box$ Lemma 7.6. Under the hypotheses of Theorem 5.2, the limiting function $c(\boldsymbol{x},t)$ belongs to $\mathbb{H}^{2+\gamma,\, (2+\gamma)/2}(\overline{\Omega}_T)$ and the following estimates hold: The estimate (7.34) is obtained by letting $\varepsilon\to 0$ in (5.5). The estimate (7.33) follows from (7.34) and the local estimates for the classical solutions of (4.13) (see [25], Ch. IV, § 10, Theorem 10.1). The existence of classical solutions of the diffusion-convection problem $\mathbb{H}(r)$ follows from (7.33) in view of Theorems 5.2 and 5.3 in [25], Ch. IV, § 5. 7.5. Properties of the operators $\mathbb{F}^v(r)$, $\mathbb{F}^p(r)$ and $\mathbb{F}^c(r)$By construction, the operators $\mathbb{F}^v(r)$, $\mathbb{F}^p(r)$ and $\mathbb{F}^c(r)$ map the set $\mathfrak{M}_T$ to the spaces $\mathbb{H}^{\gamma,\gamma/2}(\overline{\Omega}_T)$, $\mathbb{H}^{\gamma,\gamma/2}(\overline{\Omega}_T)$ and $\mathbb{H}^{2+\gamma,\,(2+\gamma)/2}(\overline{\Omega}_T)$ respectively. Lemma 7.7. Under the hypotheses of Theorem 5.2, the operators $\mathbb{F}^v(r)$, $\mathbb{F}^p(r)$ and $\mathbb{F}^c(r)$ are Lipschitz continuous: Proof. The proof is standard and expresses the well-known fact of continuous dependence of solutions of linear elliptic and parabolic equations on the coefficients. $\Box$
§ 8. Proof of Theorem 5.3We easily see that the operator $\mathbb{F}(r)$ (defined in (4.22)) is Lipschitz continuous. Moreover, for small time intervals $(0,T_1)$, it is a contraction of $\mathfrak{M}_T$ to itself. Indeed, put
$$
\begin{equation*}
R_i(\boldsymbol{x},t)=\mathbb{F}(r_i)= \int_0^tc_i(\boldsymbol{x},\tau)\, d\tau,\qquad i=1,2,
\end{equation*}
\notag
$$
where $c_i=\mathbb{F}^c(r_i)$. We have
$$
\begin{equation*}
\begin{gathered} \, 0\leqslant \mathbb{F}(r)(\boldsymbol{x},t)\leqslant T_1 c^*,\qquad |\mathbb{F}(r)|^{(2+\gamma)}_{\Omega_T}\leqslant T_1M(M_0), \\ |\mathbb{F}(r_1) - \mathbb{F}(r_{2)}|^{(2+\gamma)}_{\Omega_T}\leqslant T_1M(M_0) |r_1 - r_2|^{(2+\gamma)}_{\Omega_T}. \end{gathered}
\end{equation*}
\notag
$$
Hence $\mathbb{F}(r)$ is a contraction of $\mathfrak{M}_{T_1}$ to itself on every interval $(0,T_1)$ with
$$
\begin{equation*}
T_1<\min\biggl\{\frac{1}{2c^*},\,\frac{M(M_0)}{2}\biggr\}.
\end{equation*}
\notag
$$
Banach’s theorem ([20], Ch. II, § 4, Theorem 1) guarantees the existence of a unique fixed point $r^*(\boldsymbol{x},t)$ in $\mathfrak{M}_{T_1}$. Hence Theorem 5.3 holds on the interval $(0,T_1)$. We now put $r_1(\boldsymbol{x})=r^*(\boldsymbol{x},T_1)$ and consider Problem $\mathbb{B}^{\varepsilon}(r)$ on the interval $(T_1,T)$ with $r_0(\boldsymbol{x})$ replaced by $r_1(\boldsymbol{x})$ and $\overline{r}(\boldsymbol{x},t)$ replaced by the function $\overline{r}_1(\boldsymbol{x},t)= \max\{0,\,r_1(\boldsymbol{x})-r(\boldsymbol{x},t)\}$. We also put $\Omega_{(T_1,T)}=\Omega\times(T_1,T)$ and
$$
\begin{equation*}
\begin{aligned} \, \mathfrak{M}_{(T_1,T)} &=\biggl\{r\in \mathbb{H}^{2+\gamma,\, (2+\gamma)/2}(\overline{\Omega}_{(T_1,T)},\, r(\boldsymbol{x},T_1)=0,\, 0\leqslant r(\boldsymbol{x},t)\leqslant\frac{1}{2}, \\ &\qquad\qquad-\theta\leqslant \frac{\partial r_1}{\partial t}(\boldsymbol{x},t)\leqslant 0,\, |r_1|^{(2+\gamma)}_{\Omega_{(T_1,T)}}\leqslant M_0\biggr\}. \end{aligned}
\end{equation*}
\notag
$$
As above, we define solutions $\{\boldsymbol{v}_1,p_1,c_1\}$ and $\{r^*_1,\boldsymbol{v}_1,p_1,c_1\}$ of Problems $\mathbb{H}(r)$ and $\mathbb{H}$ on the intervals $(T_1,T)$ and $(T_1,T_2)$ respectively. Repeating the process. we obtain time intervals $(T_{k},T_{k+1})$, $k=1,2,3,\dots$, and a function $r^*$ which is equal to $r_{k}^*$ on $(T_{k},T_{k+1})$, $k=1,2,\dots$ . These functions are clearly solutions of Problem $\mathbb{H}$ on the intervals $(0,T_{k})$. If $\lim_{k\to\infty} T_{k}=\infty$, then the theorem is proved. If $\lim_{k\to\infty} T_{k}=T^*<\infty$ and $r^*(\boldsymbol{x},T^*)$ is non-zero on an open subset of $\overline{\Omega}$, then the estimates (obtained above) for solutions of Problem $\mathbb{H}$ enable us to calculate the limits of solutions as $t\to T^*$ and then find a solution of Problem $\mathbb{H}$ on the interval $(T^*,T^*+\delta)$ for some small $\delta>0$. This contradicts our assumption. Thus, the process terminates only if $r^*(\boldsymbol{x},T^*)=0$ on $\Omega$. For $t>T^*$ the liquid will completely fill the domain $\Omega$, and its motion will be described by the Stokes equations. This completes the proof of Theorem 5.3. 
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\Bibitem{Mei22} |
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