Homogenization of Smoluchowski Equations in Thin Heterogeneous Porous Domains

The Smoluchowski equation modeling Alzheimzer’s disease (AD) is a system of partial differential equations that describes the evolving densities of diffusing particles that are prone to coagulate in pairs. Recently, the important role of Smoluchowski equation in modeling the evolution of AD at different scales has been investigated in [1,12,14,29] in which the authors presented a mathematical model for the aggregation and diffusion of $\beta$-amyloid (A$\beta$) in the brain affected by AD at a microscopic scale (the size of a single neuron) and at the early stage of the disease when small amyloid fibrils are free to move and to coalesce. We also refer to [2,3,10,15,22,23,24,26] for some other works in the same direction. In the model proposed in [12], a very small portion of the cerebral tissue is described by a bounded smooth region $\mathrm{\Omega }\subset {\mathbb{R}}^3$ which is perforated by removing from it a set of periodically distributed holes of size $\epsilon$ (the neurons). Moreover the production of A$\beta$ in monomeric form at the level of neuron membranes is modeled by a non homogeneous Neumann condition on the boundary of the porosities.

In the current work, we consider the model stated in [12], but this time in a thin porous layer. This is motivated by the fact that Alzheimer’s disease particularly affects the cerebral cortex (responsible for language and information processing) and hippocampus (essential for memory), which represent very thin layers of brain tissue and contain thousands millions of neurons. Here we describe a very small layer of the brain tissue by a highly heterogeneous thin porous layer in which the heterogeneities are due to the number of millions of neurons that the brain tissue can contain. To be more precise, our model problem at the micro level is stated below.

Let $\mathrm{\Omega }$ be a bounded open Lipschitz connected subset in ${\mathbb{R}}^2$. For $0<\epsilon <1$ be freely fixed, we set We denote by $Z=Y\times I$ the reference layer cell, where $Y={(0,1)}^2$ and $I=(-1,1)$. Let $Z_f\subset Z$ be a compact set in Z with smooth boundary, which represents a generic neuron, and let $Z_s=Z\setminus Z_f$ be the supporting cerebral tissue (often call the solid part in the literature of porous media). Next, let $K_{\epsilon }=\{k\in {\mathbb{Z}}^2\times \left\{0\right\}:\epsilon (k+Z)\subset {\mathrm{\Omega }}_{\epsilon }\}$, and set $T^{\epsilon }={\cup }_{k\in K_{\epsilon }}\epsilon (k+Z_f)$. We define the thin porous layer by The boundary of ${\mathrm{\Omega }}^{\epsilon }$ is divided into two parts: the outer boundary ${\partial }_D{\mathrm{\Omega }}^{\epsilon }=\partial {\mathrm{\Omega }}_{\epsilon }$ and the inner boundary ${\mathrm{\Gamma }}^{\epsilon }=\partial T^{\epsilon }$. We also denote by $\mathrm{\Gamma }=\partial Z_f$, so that ${\mathrm{\Gamma }}^{\epsilon }={\cup }_{k\in K_{\epsilon }}\epsilon (k+\mathrm{\Gamma })$. Finally we denote by $\nu$ the outward unit normal to ${\mathrm{\Gamma }}^{\epsilon }$. We assume that ${\mathrm{\Omega }}^{\epsilon }$ is connected and that $\left|Z_s\right|>0$, where $\left|Z_s\right|$ stands for the Lebesgue measure of $Z_s$ in ${\mathbb{R}}^3$. The $\epsilon$-model reads as follows: for $m=1$, $u_1^{\epsilon }$ solves the PDE for $1<m<M$, $u_m^{\epsilon }$ solves the PDE and for $m=M$, $u_M^{\epsilon }$ solves the equation where We assume that:

H1. the coefficients $a_{i,j}$ are positive constants and satisfy $a_{i,j}=a_{j,i}$ ($1\le i,j\le M$) with $a_{M,M}=0$, and that the diffusion coefficients $d_i$ are positive constants that become smaller as j is large.

H2. The function ${\psi }^{\epsilon }$ is defined by ${\psi }^{\epsilon }(t,x)=\psi (t,\overline{x},\frac{x}{\epsilon })$ ($(t,x)\in Q_{\epsilon }$), where $\psi \in {\mathcal{C}}^1([0,T];{\mathcal{C}}^1(\overline{\mathrm{\Omega }};$${\mathcal{C}}_{per}^1(Y;{\mathcal{C}}^1\left(I\right))\left)\right)$ with $0\le \psi \le 1$ and $\psi (0,\overline{x},y)=0$ for $(\overline{x},y)\in \mathrm{\Omega }\times Z$.

In (H2), ${\mathcal{C}}_{per}^1(Y;{\mathcal{C}}^1\left(I\right))$ denotes the space of functions in ${\mathcal{C}}^1({\mathbb{R}}^2;{\mathcal{C}}^1\left(I\right))$ that are Y-periodic. In (1)-(3), ∇ stands for the usual gradient operator while $\mathrm{div}$ denotes the divergence operator with respect to the variable x; T is a positive number representing the final time. The unknowns are the vectors value functions ${\mathit{u}}^{\epsilon }:Q_{\epsilon }\to {\mathbb{R}}^M$, ${\mathit{u}}^{\epsilon }=(u_1^{\epsilon },...,u_M^{\epsilon })$ where the coordinate $u_m^{\epsilon }\ge 0$ ($1\le m<M$) stands for the concentration of m-clusters, that is clusters made of m identical elementary particles, while $u_M^{\epsilon }$ takes into account aggregation of more than $M-1$ monomers. It is worth noting that the meaning of $u_M^{\epsilon }$ is different from that of $u_m^{\epsilon }$ ($m<M$) as it aims at describing the sum of densities of all the large assemblies. It is assumed that the large assemblies exhibit all the same coagulation properties and do not coagulate with each other. We also assume that the only reaction allowing clusters to form large clusters is a binary coagulation mechanism, while the movement of clusters leading to aggregation arises only from a diffusion process described by the constant diffusion coefficient $d_m$ ($1\le m\le M$). The kinetic coefficient $a_{i,j}$ arises from a reaction in which an $(i+j)$-cluster is formed from an i-cluster and a j-cluster. Therefore, they can be interpreted as coagulation rates. Finally, $f_m^{\epsilon }$ ($1<m<M$) accounts for the formation of m-clusters by coalescence of smaller clusters and $g_{\epsilon }$ accounts for the formation of a large clusters by coalescence of others large one that have the same coagulation properties.

Our main aim in this work is to investigate the limiting behaviour as $\epsilon \to 0$, of the solution ${\mathit{u}}^{\epsilon }$ to (1)-(3) under the assumptions (H1)-(H2). This falls within the scope of the homogenization theory in thin porous domains.

There is a huge literature on homogenization in fixed or porous media. A few works deal with the homogenization theory in thin heterogeneous domains; see e.g. [4,5,6,8,9,11,16,17,18,21,25]. As for the homogenization in thin heterogeneous porous media, very few results are known up to now. We may cite [4,5,6,8,11]. Concerning the Smoluchowski equation as stated in this work, to the best of our knowledge, the only work dealing with its homogenization is the paper [12] in which the considered domain is a uniformly perforated one that is not thin. Because our domain is thin and porous, the homogenization process is not an easy task. Indeed, we make use of the partial mean integral operator $M_{\epsilon }$ (see below for its definition) associated to the extension operator, while in [12], even the extension operator is not used. So, the main novelty in our work arises from the fact that the domain ${\mathrm{\Omega }}^{\epsilon }$ is a thin heterogeneous porous layer. This leads to a dimension reduction problem in the limit as shown here below in the main result, which reads as follows.

The partial mean integral $M_{\epsilon }$ considered in Theorem 1 is defined, for a function $\varphi$ by

The system (8)-(10) is the upscaled model arising from the $\epsilon$-model (1)-(3). It is posed in a 2 dimensions space, leading to an expected dimension reduction problem as it is usually the case for the homogenization theory in thin domains. Moreover the information given on the microscale by the Neumann boundary condition in (1) is transferred (in the limit) into the source term in the leading equation in (8), so that, in the case of (1), the limiting equation does not have the same form as the original equation posed in the $\epsilon$-model. For (9) and (10), apart from the diffusion term, they are similar to the $\epsilon$-equations in (2) and (3).

The rest of the paper is organized as follows. In Section 2, we investigate the well posedness of (1)-(3) and provide useful uniform estimates. Section 3 deals with the treatment of the concept of two-scale convergence for thin heterogeneous domains. We prove therein some compactness results that will be used in the homogenization process. With the help of the results obtained in Section 3, we pass to the limit in (1)-(3) in Section 4 where we prove the main result, viz. Theorem 1.

The current section deals with the existence and uniqueness of the solution to (1)-(3), together with some uniform estimates that will be useful in the sequel. The following result holds true.

The following result whose proof can be found in [19, Theorem 3] will be useful in the sequel.

Based on Lemma 3, we define the extension $E_{\epsilon }v$ of a function $v\in L^2(0,T;H^1\left({\mathrm{\Omega }}^{\epsilon }\right))$ to $L^2(0,T;H^1\left({\mathrm{\Omega }}_{\epsilon }\right))$ as follows:Then accounting of Lemma 3 and Theorem 2, we have where $C>0$ is independent of $\epsilon$ and We also need an estimate on $\partial u_m^{\epsilon }/\partial t$ in $L^2\left({\mathrm{\Omega }}_{\epsilon }^T\right)$. To that end, we procced as in [20] and consider the restriction operator $R_{\epsilon }:L^2\left({\mathrm{\Omega }}_{\epsilon }\right)\to L^2\left({\mathrm{\Omega }}^{\epsilon }\right)$, $R_{\epsilon }v={\left.v\right|}_{{\mathrm{\Omega }}^{\epsilon }}$ (the restriction of v to ${\mathrm{\Omega }}^{\epsilon }$). Then it is a fact that $R_{\epsilon }$ is a bounded linear operator as Now, if $R^{*}:L^2\left({\mathrm{\Omega }}^{\epsilon }\right)\to L^2\left({\mathrm{\Omega }}_{\epsilon }\right)$ denotes the adjoint operator of $R_{\epsilon }$, then for $v\in L^2(0,T;L^2\left({\mathrm{\Omega }}^{\epsilon }\right))=L^2\left(Q_{\epsilon }\right)$, we define $R_{\epsilon }^{*}v$ as follows: and we have for all $u\in L^2\left(Q_{\epsilon }\right)$ and $v\in L^2\left({\mathrm{\Omega }}_{\epsilon }^T\right)$. It is therefore easy to see that $R_{\epsilon }^{*}v={\chi }_{{\mathrm{\Omega }}^{\epsilon }}v$ for all $v\in L^2\left(Q_{\epsilon }\right)$, or equivalently

The two-scale convergence for thin heterogeneous domains has been introduced in [25] and extended to thin porous surfaces in [8,19]. The notations used in this section are the same as in the previous ones. Especially, the domain ${\mathrm{\Omega }}_{\epsilon }$ is defined as above, that is, ${\mathrm{\Omega }}_{\epsilon }=\mathrm{\Omega }\times (-\epsilon ,\epsilon )$. When $\epsilon \to 0$, ${\mathrm{\Omega }}_{\epsilon }$ shrinks to the "interface" ${\mathrm{\Omega }}_0=\mathrm{\Omega }\times \left\{0\right\}\equiv \mathrm{\Omega }$. We know that $Q_{\epsilon }=(0,T)\times {\mathrm{\Omega }}^{\epsilon }$ and ${\mathrm{\Omega }}_{\epsilon }^T=(0,T)\times {\mathrm{\Omega }}_{\epsilon }$, and we set $Q=(0,T)\times {\mathrm{\Omega }}_0$, $I=(-1,1)$, $Y={(0,1)}^2$ and finally $Z=Y\times I$. Let $1\le p<\infty$; by $L_{per}^p(Y;L^p\left(I\right))$ we denote the space of functions in $L_{loc}^p({\mathbb{R}}^2;L^p\left(I\right))$ that are Y-periodic. Accordingly we define $W_{per}^{1,p}(Y;W^{1,p}\left(I\right))$ as the subspace of $W_{loc}^{1,p}(Y;W^{1,p}\left(I\right))$ made of periodic Y-periodic functions, and we set which is a Banach space equipped with the norm

Any x in ${\mathbb{R}}^3$ writes $(\overline{x},x_3)$ or $(\overline{x},\zeta )$ where $\overline{x}=(x_1,x_2)$. We identify ${\mathrm{\Omega }}_0$ with $\mathrm{\Omega }$ so that the generic element in ${\mathrm{\Omega }}_0$ is also denoted by $\overline{x}$ instead of $(\overline{x},0)$.

We are now able to define the two-scale convergence for thin heterogeneous domains and for thin boundaries.

We start with the following important result that should be used in the sequel; see [7, Lemma 3.2.3] for the proof.

Throughout the work, the letter E will stand for any ordinary sequence ${\left({\epsilon }_n\right)}_{n\ge 1}$ with $0<{\epsilon }_n\le 1$ and ${\epsilon }_n\to 0$ when $n\to \infty$. The generic term of E will be merely denote by $\epsilon$ and $\epsilon \to 0$ will mean ${\epsilon }_n\to 0$ as $n\to \infty$. This being so, we have the following compactness results.