Abstract
In a thin heterogeneous porous layer, we carry out a multiscale analysis of Smoluchowski’s discrete diffusion–coagulation equations describing the evolution density of diffusing particles that are subject to coagulation in pairs. Assuming that the thin heterogeneous layer is made up of microstructures that are uniformly distributed inside, we obtain in the limit an upscaled model in the lower space dimension. We also prove a corrector-type result very useful in numerical computations. In view of the thin structure of the domain, we appeal to a concept of two-scale convergence adapted to thin heterogeneous media to achieve our goal.
MSC:
35B27; 35F61; 92B05
1. Introduction and the Main Results
The use of the Smoluchowski equation has proved very efficient in modeling several natural and physical phenomena in chemistry, astrophysics, aerosol science, physics, engineering and biological sciences, just to cite a few. Some applications arise in the modeling of 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 red blood cell aggregation. In this work, we are particularly interested in its application to the aggregation and diffusion of particles.
More precisely, we are concerned with the application of the Smoluchowski equation in the modeling of Alzheimer’s disease (AD), as it is a system of partial differential equations aimed at describing the evolving densities of diffusing particles subject to coagulation in pairs. Recently, the crucial role of the Smoluchowski equations in the multiscale modeling of the evolution of AD at different scales has been considered in [,,,], where the authors proposed a suitable mathematical model for the aggregation and diffusion of -amyloid (A) in the brain affected by AD at the micro-scale (that is, at the size of a single neuron) and at the primary step of the disease when small amyloid fibrils are free to move and merge. We also refer to [,,,] for some other works in the same direction. In the model considered in [], a tiny part of cerebral tissue is viewed as a bounded domain , which is perforated by removing from it a set of periodically distributed holes of size (the neurons). Moreover, the production of A in monomeric form at the level of neuron membranes is modeled using a non-homogeneous Neumann condition on the boundary of the porosities.
In the current work, we consider the model stated in [] 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 and millions of neurons. Here, we describe a very small layer of brain tissue by using 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 be a bounded open Lipschitz connected subset in . For to be freely fixed, we set
We denote by the reference layer cell, where and . Let be a compact set in Z with a smooth boundary, which represents a generic neuron, and let be the supporting cerebral tissue (often called the solid part in the literature of porous media).
Let us set a notation that is used throughout this work. Let . For any set and any (with denoting the integers), we set
With this in mind, let , and set . We define the thin porous layer by
The boundary of is divided into two parts: the outer boundary and the inner boundary . We also denote by so that . Finally we denote by the outward unit normal to . We assume that is connected and that , where stands for the Lebesgue measure of in . The -model reads as follows: for , solves the PDE
for , solves the PDE
and for , solves the equation
where
We assume the following:
Hypothesis 1 (H1).
The coefficients are positive constants and satisfy () with , and the diffusion coefficients are positive constants that become smaller as j becomes large;
Hypothesis 2 (H2).
The function is defined by (), where with and for .
In (H2), denotes the space of functions in 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 vector value functions , , where the coordinate () stands for the concentration of m-clusters, that is, the clusters made of m identical elementary particles, while takes into account the aggregation of more than monomers. It is worth noting that the meaning of is different from that of (), as it aims to describe 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 (). The kinetic coefficient arises from a reaction in which an -cluster is formed from an i-cluster and a j-cluster. Therefore, they can be interpreted as coagulation rates. Finally, () represents the formation of m-clusters via the coalescence of smaller clusters, and accounts for the formation of large clusters via the coalescence of other large ones that have the same coagulation properties.
Our main aim in this work is to investigate the limiting behavior as of the solution to (1)–(3) under the assumptions (H1)–(H2). This falls within the scope of a 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 numerically model such features and capture these multiscale phenomena as correct as possible, 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 tremendous development during the past 50 years is the theory of homogenization. Roughly speaking, homogenization consists of replacing the generally complicated study of heterogeneous and composite phenomena, often modeled using (nonlinear) partial differential equations (PDEs) with variable coefficients, by the study of equivalent homogeneous phenomena with the same overall properties but modeled using PDEs with non-oscillating coefficients, which is ideal for numerical analyses, interpretation and predictions, hence the important role of this step. Homogenization offers a rigorous mathematical framework allowing for the modeling and analysis of composites in various environments. This is especially the case when the environment is represented by a domain that is the union (or the complement of the union) of subdomains of a very small size, say, a domain containing infinitely many holes such as the one under consideration in this work. That is why the macroscopic model that is 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., [,,,,,,]. As for homogenization in thin heterogeneous porous media, very few results are known up to now. We may cite [,,,,]. Concerning the Smoluchowski equation as stated in this work, to the best of our knowledge, the only work dealing with its homogenization is the study in [], in which the considered domain is a uniformly perforated one that is not thin. Our contribution in this work is twofold: (1) The domain 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 (see below for its definition) associated with the extension operator, while in [], even the extension operator is not used. (2) We prove in Theorem 2 a corrector-type result allowing us to approximate each by a function of the form , where the functions and do not depend on . We summarize our main results below.
Theorem 1.
Assume that (H1)–(H2) hold. For any , let be the unique solution of (1)–(3) in the class , (). Let also and respectively denote the partial mean integral operator and the extension operator defined by (37) (see Section 3) and in Lemma 1 (see Section 2). Then, as , one has, for any ,
where is the unique solution of system (8)–(10) below:
If ,
and
Moreover, and is such that
In (8)–(10), n is the outward unit normal to and the matrix , where is the identity matrix and , with being the unique solution (up to the addition of function such that in ) in of the cell problem
where, here, ν stands for the outward unit normal to Γ, and is the ith vector of the canonical basis in ; the functions and θ are respectively defined by , and (the Lebesgue measure of in ).
The partial mean integral considered in Theorem 1 is defined, for function , by
System (8)–(10) is the upscaled model arising from the -model (1)–(3). It is posed in a two-dimensional space, leading to an expected dimension reduction problem, as usually is the case for the homogenization theory in thin domains. Moreover, the Neumann boundary behavior in (1) now plays 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 -model. For (9) and (10), apart from the diffusion term, they are similar to the -equations in (2) and (3).
Now, let () and () be as in Theorem 1. We set
where . We have , where stands for the space of u functions in that are Y-periodic and satisfy .
With this in mind, the second main result is a corrector-type result and reads as follows:
Theorem 2.
For each , assume that defined by (13) belongs to , where is Y-periodic and . Then, as , one has
where for .
The result in Theorem 2 allows us to approximate in by function of the form for . Theorem 2 is new in the literature of the homogenization of the Smoluchowski equation and is very important as far as the quantitative homogenization theory of such kind of equations is concerned.
The plan of this work is 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 the two-scale convergence of thin heterogeneous domains. We prove therein some compactness results that are 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 we close the work with a conclusion.
2. Well Posedness and Uniform Estimates
The current section deals with the existence and uniqueness of the solution to (1)–(3), along with some useful a priori estimates. We begin with the following theorem:
Theorem 3.
Proof.
The well posedness of (1)–(3) has been addressed in [,,,]. We are concerned here only with the uniform estimates (15)–(17), with estimate (18) being a classical result arising from the trace result. We just emphasize that, since ( stands for the Lebesgue measure of ), no scaling is needed in the left-hand side of (18). Now, for (15), we follow exactly the same lines of reasoning as in [] to obtain it. Both (16) and (17) remain to be checked. We first consider (16). We distinguish the cases and .
We start with . By multiplying (1) by and integrating over , followed by the use the divergence theorem, we obtain
where the last inequality above stems from Hölder’s and Young’s inequalities. We use a well-known trace inequality to deduce the existence of a positive constant independent of such that
Therefore, by integrating (19) over () and taking into account (18) and (20), we are led to
We therefore infer the boundedness of in associated with (21) wherein there exists such that (16) holds for and for all , where is chosen such that , that is, .
For , we proceed as for and multiply (2) by and integrate over ; then, one obtains
By integrating over for , we obtain
By using (15), we obtain at once
Finally, the proof of (16) for is obtained exactly as the one for the case mutatis mutandis (replace with ).
Let us now prove (17). We proceed as above by distinguishing three cases.
For , we multiply (1) by and use (1)–(1) to obtain
But
Thus,
By integrating (22) over and using the boundedness property (15), we obtain after integration by parts
where we use the fact that . Now, we use inequality (20); then, (23) becomes
It follows that
where, in (24), we took advantage of (15) and (16). Hence, by choosing to be sufficiently small so that , we obtain (17) for .
The proof of (17) in the case when follows the same lines of reasoning as above, but it is much easier. It is therefore left to the reader. This completes the proof. □
The following result whose proof can be found in Theorem 3 in [] will be useful in the sequel.
Lemma 1.
There exists a bounded linear operator such that, for all , in and
and
for a positive constant independent of both ε and v.
By virtue of Lemma 1, we may define the extension operator from into via the following statement:for , we have
Then, on account of Lemma 1 and Theorem 3, we have
where is independent of and
We also need an estimate of in . To this end, we proceed as in [] and consider the restriction operator , (the restriction of v to ). Then, it is a fact that is a bounded linear operator as
Now, if denotes the adjoint operator of , then, for , we define by
Then, one has
for all and . It is therefore easy to see that for all , or equivalently
where stands for the characteristic function of in .
Lemma 2.
Let the assumptions of Theorem 3 hold. It holds that
where is independent of ε, and is defined in Theorem 3.
Proof.
First, we have , where . Thus, it is sufficient to show that
So, let ; then,
Whence the result. □
3. Two-Scale Convergence of Thin Heterogeneous Domains
The two-scale convergence of thin heterogeneous domains has been introduced in [] and extended to thin porous surfaces in [,]. The notations used in this section are the same as in the previous ones. Specifically, the domain is defined as above, that is, . When , shrinks to the “interface” . We know that and , and we set , , and finally . Let ; by , we denote the space of functions in that are Y-periodic. Accordingly, we define as the subspace of made of Y-periodic functions, and we set
which is a Banach space equipped with the norm
Any x in writes or , where . We identify with so that the generic element in is also denoted by instead of .
We are now able to define the two-scale convergence of thin heterogeneous domains and thin boundaries.
Definition 1.
(a) The sequence () is
- (i)
- Weakly two-scale convergent in to if whenever , one has
- (ii)
- Strongly two-scale convergent in towards if, as , one has in -weak and(b) The sequence in is weakly two-scale convergent in towards if, whenever , one has
Remark 1.
We start with the following important result that should be used in the sequel; see Lemma 3.2.3 in [] for the proof.
Lemma 3.
Let , which is Y-periodic in . Then, by letting for , we have
- (i)
- ;
- (ii)
Throughout this work, the letter E stands for any ordinary sequence with and when . The generic term of E is merely denoted by , and means as . This being so, we have the following compactness results.
Theorem 4.
(i) Let be a sequence in such that
where C is a positive constant independent of ε. Then, up to a subsequence of E, the sequence weakly two-scale converges in to some .
(ii) Let be a sequence in such that
with being independent of ε. Then, we may find a subsequence of E such that the sequence weakly two-scale converges in towards some function .
In Theorem 4 above, the proof of part (i) can be found in [], while the proof of part (ii) can be found in [] (see also [,]).
Theorem 5.
Let be a sequence in () such that
where is independent of ε. Then, up to a subsequence extracted from E, we may find a vector function with and such that, when , we have
and
For the proof of Theorem 5, we refer to [].
The following result is sharper than its homologue in Theorem 5.
Theorem 6.
Let be a sequence in such that
where C is a positive constant independent of ε. Finally, suppose that the embedding is compact. Then, up to a subsequence of E, there is a vector function such that, as ,
and
Proof.
First, owing to Theorem 5, we derive the existence of a subsequence of E and of a vector function such that, as ,
and
Now, (34) remains to be proved. To this end, we set
Then, we easily see that with
Then, from (38), we derive the existence of a subsequence of still denoted by and of the function such that, as ,
We recall that (39) stems from the compactness of the continuous embedding .
The next result and its corollary are proved exactly as their homologues in Theorem 6 and Corollary 5 in [] (see also []).
Theorem 7.
Let and be such that . Suppose that weakly two-scale converges in towards and strongly two-scale converges in towards . Then, is weakly two-scale convergent in to .
Corollary 1.
Assume the sequences in and in (with , ) satisfy the following:
- (i)
- in -weak ;
- (ii)
- in -strong ;
- (iii)
- is bounded in .
Then, in -weak .
4. Derivation of the Homogenized Problem: Proofs of the Main Results
4.1. Preliminary Results
In this subsection, we aim to provide further important convergence results that will be very useful in the sequel. In that order, it is to be noted that can alternatively be defined as follows: , where with and . We set , a periodic repetition of set . We denote by the characteristic function of in : . Then, it holds that
so that for .
Lemma 4.
Let be a sequence in ( a real number), which is weakly two-scale convergent in to . Then, as ,
If further two-scale convergence is strong, then (41) holds in the strong two-scale sense.
Proof.
Set for . Then, since in -weak , it holds that (with being independent of ) so that . Hence, up to a subsequence, in in the usual classical two-scale weak sense, where . Next, let . By passing to the limit (in the subsequence determined above) in the obvious equality
we obtain at once .
This being so, by choosing f as above, one has
Owing to the usual two-scale concept, we obtain, as ,
where, in (42), we used the fact that as proven above. This concludes the proof. □
The following result will be crucial in the homogenization process. From now on, we set , the characteristic function of in Z.
Proposition 1.
4.2. Passage to the Limit
Assume that the functions and are as in Proposition 1. Let and , and define
We use as a test function in the variational form of (1)–(3):
For ,
and
Let us first deal with (50). We note that it is equivalent to
We have that
Thus, we may apply Proposition 1 to proceed to the passage to the limit in the first two terms of the left-hand side of (53), using as a test function in the two-scale concept. Concerning the right-hand side of (53), we use Lemma 3 to pass to the limit therein. We end up with the last term on the left-hand side, where the limit passage therein is more involved. Indeed, we use there the strong two-scale convergence of towards associated with the weak two-scale convergence of () towards to obtain from Corollary 1 that, for , we have, as ,
Therefore, by using in that term the test function and taking into account all the processes described above after (53), we are led, as in (53), to
We use the same process as for (53) to pass to the limit in (51) and in (52), and we obtain the following:
- For ,
We have proved the following result.
Our next goal is to derive the system whose solution is . To this end, we start by uncoupling Equations (55)–(57). We first consider (55), and we see that it is equivalent to the following system consisting of (58) and (59) below:
Let us first consider Equation (58) and choose therein under the form with and ; then, (58) becomes
To solve (60), we instead consider the variation problem
where () denotes the jth vector of the canonical basis of . Then, (61) is equivalent to the cell problem
where stands for the outward unit normal to . It is an easy task to see that (62) possesses a solution in the space
that is unique up to the addition of a function such that in . Now, by multiplying (61) by () and summing up the resulting equations, and then comparing the latter sum with (60), the following is yielded at once:
where .
Next, by going back to (59) and replacing with the expression obtained in (63), we obtain
where is the identity matrix.
This being so, we set
Then, A is a symmetric positive definite matrix. Indeed, it is a fact that the entries of A have the form
this stems from (61), where we show that it is still valid for and then choose therein . With the above notations in (65), we see that (64) is equivalent to the problem
Proceeding as we did for (55), we easily show that (56) and (57) are equivalent to the variational formulations of the following PDEs:
4.3. Proof of Theorem 1
The proof of (5)–(7) follows easily from (47)–(49) associated with the properties of operator . The fact that solves (8)–(10) has been shown here above in Section 4.2. Now, if we proceed as in [] (see also []), we obtain the well posedness of (8)–(10) in the space , and, specifically, (11) holds true. Indeed, if we set , where
Then, F satisfies the assumptions of the appendix in []. Hence, Theorems 7.1 and 7.2 of [] readily ensure the existence and uniqueness of the solution of (8)–(10) as claimed above. Finally, the fact that the whole sequence converges towards follows from the uniqueness of the solution of (8)–(10). This concludes the proof.
4.4. Proof of Theorem 2
First of all, we recall that for . This being so, for to be freely fixed, let . Then, . Assuming , the functions , and belong to so that they can be used as test functions in the definition of the two-scale convergence (see Definition 1).
This being so, let us first consider the case .
We have
Thus, by taking into account (43) (or (47)), proving Theorem 2 amounts to showing that as . So, we have
where in the series of equalities above, we omitted in the integrals just for the simplification of the presentation. We use and as test functions to obtain at once
and
With regard to , one has
By appealing to (54) and using once more the strong two-scale convergence of towards , we obtain
Also, the strong two-scale convergence of associated with the weak two-scale convergence of gives, owing to Corollary 1,
Now, for the last term on the right-hand side of (72), we first notice that, from the well-known trace inequality
we have from (15) and (16)
where is independent of . It follows from part (ii) of Theorem 4 that (up to a subsequence) the trace of on two-scale converges in , and its two-scale limit can be easily identified (by integration by parts) with the trace of on , i.e.,
Thus, by using as a test function, we obtain, up to a subsequence,
Now, in view of the uniqueness of , the convergence result in (77) holds with the entire sequence .
By collecting (73), (74) and (77), we obtain
Now, if we take as a test function in the variational form of (66) and account for (78), we see that
The proof in the case is easier and follows the same steps as in the case . Theorem 2 is therefore proved.
5. Conclusions
In this work, we provided a qualitative multiscale analysis of a micro-model of Smoluchowski equations in thin heterogeneous domains. Starting from a three-dimensional problem, we proved that the upscaled equation is posed on a two-dimensional space, leading to a dimension reduction problem. We also addressed an approximation issue by proving a corrector-type result, showing that the solution can be approximated by the function in where and solve equations that are independent of . This is very useful in numerical computations and opens the door to the quantitative homogenization of (1), which aims to find the rate of convergence in the approximation of by .
Author Contributions
Writing—original draft, J.L.W.; Writing—review & editing, R.G.N. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Acknowledgments
The authors thank the referees for their helpful comments and suggestions.
Conflicts of Interest
The authors declare no conflict of interest.
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