The use of Smoluchowski equation has proved very efficient in modelling several natural and physical phenomena in Chemistry, in Astrophysics, in Aerosol science, in Physics, in Engineering and in Biological sciences, just to cite a few. Some applications arise in the modeling of the polymerization in Chemistry, the motion of a system of particles that are suspended in a gas, the behavior of fuel mixtures in engines (in Engineering science), the formation of stars and planets (in Physics) and in the modeling of red blood cell aggregation. In this work, we are particularly interested in its application to aggregation and diffusion of particles.

More precisely, we are concerned with Smoluchowski equation modelling Alzheimzer’s disease (AD) which is a system of partial differential equations aiming at describing the evolving densities of diffusing particles subject to coagulate in pairs. Recently, the crucial role of Smoluchowski equations in the multiscale modeling of the evolution of AD at different scales has been considered in [1,2,3,4] where the authors proposed a suitable mathematical model for the aggregation and diffusion of $\beta$-amyloid (A$\beta$) in the brain affected by AD at a microscale (that is, at the size of a single neuron) and at primary step of the disease when small amyloid fibrils are free to move and merge. We also refer to [5,6,7,8] for some other works in the same direction. In the model considered in [2], a tiny part of the cerebral tissue is viewed as a bounded domain $\Omega \subset {\mathbb{R}}^3$ 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 [2], 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 $\Omega$ 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).

Let us set a notation that will be used throughout the work. Let $0<\epsilon \le 1$. For any set $S\subset {\mathbb{R}}^3$ and any $k\in {\mathbb{Z}}^3$ ($\mathbb{Z}$ denoting the integers), we set With this in mind, let $K_{\epsilon }=\{k\in {\mathbb{Z}}^2\times \left\{0\right\}:Z^{\epsilon ,k}\subset {\Omega }_{\epsilon }\}$, and set $T^{\epsilon }={\cup }_{k\in K_{\epsilon }}{Z}_f^{\epsilon ,k}$. We define the thin porous layer by The boundary of ${\Omega }^{\epsilon }$ is divided into two parts: the outer boundary ${\partial }_D{\Omega }^{\epsilon }=\partial {\Omega }_{\epsilon }$ and the inner boundary ${\Gamma }^{\epsilon }=\partial T^{\epsilon }$. We also denote by $\Gamma =\partial Z_f$, so that ${\Gamma }^{\epsilon }={\cup }_{k\in K_{\epsilon }}{\Gamma }^{\epsilon ,k}$. Finally we denote by $\nu$ the outward unit normal to ${\Gamma }^{\epsilon }$. We assume that ${\Omega }^{\epsilon }$ 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:

In (H2), ${\mathcal{C}}_{per}^1(Y;{\mathcal{C}}^1\left(I\right))$ denotes the space of functions in ${\mathcal{C}}_{loc}^1({\mathbb{R}}^2;{\mathcal{C}}^1\left(I\right))$ that are Y-periodic. In (1)-(3), ∇ stands for the usual gradient operator while 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 ${\mathbf{u}}^{\epsilon }:Q_{\epsilon }\to {\mathbb{R}}^M$, ${\mathbf{u}}^{\epsilon }=(u_1^{\epsilon },...,u_M^{\epsilon })$ 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 coagulationrates. Finally, $f_m^{\epsilon }$ ($1<m<M$) represents 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 behavior as $\epsilon \to 0$, of the solution ${\mathbf{u}}^{\epsilon }$ to (1)-(3) under the assumptions (H1)-(H2). This falls within the scope of the multiscale analysis through the homogenization theory in thin porous domains.

Most structures in nature exhibit multiscale features both in space and time. In biological sciences, modeling and simulation have proven to be useful and necessary in describing and explaining many biological processes. To meet the challenge of their complexity, and in order to model numerically such features and capture as correct as possible these multiscale phenomena, mathematical modeling and theoretical concepts combined with the development of efficient algorithms and simulation tools must be emphasized and promoted. One such mathematical concept that has seen a tremendous development during the past 50 years is the theory of homogenization. Roughly speaking, homogenization consists in replacing the generally complicated study of heterogeneous and composite phenomena, often modeled by (nonlinear) partial differential equations (PDE) with variable coefficients, by the study of equivalent homogeneous phenomena having the same overall properties, but modeled by PDE with non oscillating coefficients, which is ideal for numerical analysis, interpretation and predictions. Hence the important role of this step. Homogenization offers a rigorous mathematical framework allowing the modeling and analysis of composites in various environments. This is especially the case when the environment is represented by a domain which is the union (or the complement of the union) of subdomains of very small size, say, a domain containing infinite many holes as the one under consideration in this work. That is why the macroscopic model that will be derived in this work is more relevant in practice than the microscopic one.

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. [9,10,11,12,13,14,15]. As for the homogenization in thin heterogeneous porous media, very few results are known up to now. We may cite [9,10,11,12,14]. 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 [2] in which the considered domain is a uniformly perforated one that is not thin. Our contribution in this work is twofold: 1) the domain ${\Omega }^{\epsilon }$ is a thin heterogeneous porous layer. This renders the homogenization procedure not easy to handle. Indeed, to achieve our goal in Theorem 1 below, we make use of the partial mean integral operator $M_{\epsilon }$ (see below for its definition) associated to the extension operator while in [2], even the extension operator is not used; 2) we prove in Theorem 2 a corrector-type result allowing us to approximate each $u_m^{\epsilon }$ by a function of the form $v_m^{\epsilon }(t,x)=u_m(t,\overline{x})+\epsilon u_m^1(t,\overline{x},x/\epsilon )$ where the functions $u_m$ and $u_m^1$ do not depend on $\epsilon$. We summarize our main results below.

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 Neumann boundary behavior in (1) plays now the role (in the upscaled model) of 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).

Now, let ${\omega }_i$ ($i=1,2$) and $u_m$ ($1\le m\le M$) be as in Theorem 1. We set where $\omega =({\omega }_1,{\omega }_2)$. We have that $u_m^1\in L^2\left(Q\right)\otimes H_{\#}^1(Y;H^1\left(I\right))$, where $H_{\#}^1(Y;H^1\left(I\right))$ stands for the space of functions u in $H_{loc}^1({\mathbb{R}}^2;H^1\left(I\right))$ that are Y-periodic and satisfy ${\int }_{Z_s}u\left(y\right)dy=0$.

With this in mind, the second main result is a corrector-type result and reads as follows.

The result in Theorem 2 allows us to approximate $u_m^{\epsilon }$ in $Q_{\epsilon }$ by a function $v_m^{\epsilon }$ of the form $v_m^{\epsilon }(t,x)=u_m(t,\overline{x})+u_m^1(t,\overline{x},x/\epsilon )$ for $(t,x)\in Q_{\epsilon }$. Theorem 2 is new in the literature of the homogenization of Smoluchowski equation and is very important as far as the quantitative homogenization theory of such kind of equations is concerned.

The plan of the work is as follows. We investigate in Section 2 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 first main result of the work, viz. Theorem 1. We also prove Theorem 2 in the same section, and close the work with a conclusion.

The current section deals with the existence and uniqueness of the solution to (1)-(3), alongside with some useful a priori estimates. We begin with the following theorem.

The following result whose proof can be found in [17] will be useful in the sequel.

By virtue of Lemma 1, we may define the extension operator from $L^2(0,T;H^1\left({\Omega }^{\epsilon }\right))$ into $L^2(0,T;H^1\left({\Omega }_{\epsilon }\right))$ by:For $v\in L^2(0,T;H^1\left({\Omega }^{\epsilon }\right))$ we have

Then accounting of Lemma 1 and Theorem 3, we have where $C>0$ is independent of $\epsilon$ and We also need an estimate on $\partial u_m^{\epsilon }/\partial t$ in $L^2\left({\Omega }_{\epsilon }^T\right)$. To that end, we proceed as in [18] and consider the restriction operator $R_{\epsilon }:L^2\left({\Omega }_{\epsilon }\right)\to L^2\left({\Omega }^{\epsilon }\right)$, $R_{\epsilon }v={\left.v\right|}_{{\Omega }^{\epsilon }}$ (the restriction of v to ${\Omega }^{\epsilon }$). Then it is a fact that $R_{\epsilon }$ is a bounded linear operator as Now, if $R^{*}:L^2\left({\Omega }^{\epsilon }\right)\to L^2\left({\Omega }_{\epsilon }\right)$ denotes the adjoint operator of $R_{\epsilon }$, then for $v\in L^2(0,T;L^2\left({\Omega }^{\epsilon }\right))=L^2\left(Q_{\epsilon }\right)$, we define $R_{\epsilon }^{*}v$ by: Then one has for all $u\in L^2\left(Q_{\epsilon }\right)$ and $v\in L^2\left({\Omega }_{\epsilon }^T\right)$. It is therefore easy to see that $R_{\epsilon }^{*}v={\chi }_{{\Omega }^{\epsilon }}v$ for all $v\in L^2\left(Q_{\epsilon }\right)$, or equivalently where ${\chi }_{{\Omega }^{\epsilon }}$ stands for the characteristic function of ${\Omega }^{\epsilon }$ in ${\Omega }_{\epsilon }$.

The two-scale convergence for thin heterogeneous domains has been introduced in [19] and extended to thin porous surfaces in [12,17]. The notations used in this section are the same as in the previous ones. Especially, the domain ${\Omega }_{\epsilon }$ is defined as above, that is, ${\Omega }_{\epsilon }=\Omega \times (-\epsilon ,\epsilon )$. When $\epsilon \to 0$, ${\Omega }_{\epsilon }$ shrinks to the "interface" ${\Omega }_0=\Omega \times \left\{0\right\}$. We know that $Q_{\epsilon }=(0,T)\times {\Omega }^{\epsilon }$ and ${\Omega }_{\epsilon }^T=(0,T)\times {\Omega }_{\epsilon }$, and we set $Q=(0,T)\times {\Omega }_0$, $I=(-1,1)$, $Y={(0,1)}^2$ 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 ${\Omega }_0$ with $\Omega$ so that the generic element in ${\Omega }_0$ 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.