From a Microscopic to a Macroscopic Model for Alzheimer Disease: Two-Scale Homogenization of the Smoluchowski Equation in Perforated Domains

Abstract

In this paper, we study the homogenization of a set of Smoluchowski’s discrete diffusion–coagulation equations modeling the aggregation and diffusion of $\beta$-amyloid peptide (A$\beta$), a process associated with the development of Alzheimer’s disease. In particular, we define a periodically perforated domain $\Omega _{\epsilon }$, obtained by removing from the fixed domain $\Omega$ (the cerebral tissue) infinitely many small holes of size $\epsilon$ (the neurons), which support a non-homogeneous Neumann boundary condition describing the production of A$\beta$ by the neuron membranes. Then, we prove that, when $\epsilon \rightarrow 0$, the solution of this micromodel two-scale converges to the solution of a macromodel asymptotically consistent with the original one. Indeed, the information given on the microscale by the non-homogeneous Neumann boundary condition is transferred into a source term appearing in the limiting (homogenized) equations. Furthermore, on the macroscale, the geometric structure of the perforated domain induces a correction in that the scalar diffusion coefficients defined at the microscale are replaced by tensorial quantities.

Appendices

Appendix A

Lemma 7.1

The following estimate holds: if $v\in \mathrm {Lip} (\Omega _{\epsilon })$, then

$$\begin{aligned} \Vert v \Vert ^2_{L^2(\Gamma _{\epsilon })} \le C_1 \, \bigg [ {\epsilon }^{-1} \displaystyle \int _{\Omega _{\epsilon }} \vert v \vert ^2 \, \mathrm{d}x+ \epsilon \displaystyle \int _{\Omega _{\epsilon }} \vert \nabla _x v \vert ^2 \, \mathrm{d}x \bigg ] \end{aligned}$$

(113)

where $C_1$ is a constant which does not depend on $\epsilon$.

The inequality (113) can be easily obtained from the standard trace theorem by means of a scaling argument (Allaire et al. 1996; Chiadò Piat and Piatnitski 2010; Chiadò Piat et al. 2012).

Lemma 7.2

Suppose that the domain $\Omega _{\epsilon }$ is such that assumption (8) is satisfied. Then, there exists a family of linear continuous extension operators

$$\begin{aligned} P_{\epsilon }: W^{1,p}(\Omega _{\epsilon }) \rightarrow W^{1,p}(\Omega ) \end{aligned}$$

and a constant $C >0$ independent of $\epsilon$ such that

$$\begin{aligned} P_{\epsilon } v=v \; \; \; \text {in } \Omega _{\epsilon } \end{aligned}$$

and

$$\begin{aligned}&\int _{\Omega } \vert P_{\epsilon } v \vert ^{p} \mathrm{d}x \le C \int _{\Omega _{\epsilon }} \vert v \vert ^p \mathrm{d}x \;, \end{aligned}$$

(114)

$$\begin{aligned}&\int _{\Omega } \vert \nabla (P_{\epsilon } v) \vert ^{p} \mathrm{d}x \le C \int _{\Omega _{\epsilon }} \vert \nabla v \vert ^p \mathrm{d}x \end{aligned}$$

(115)

for each $v \in W^{1,p}(\Omega _{\epsilon })$ and for any $p \in (1, +\infty )$.

For the proof of this Lemma see, for instance, (Chiadò Piat and Piatnitski 2010).

As a consequence of the existence of extension operators, one can derive the Sobolev inequalities in $W^{1,p}(\Omega _{\epsilon })$ with a constant independent of $\epsilon$.

Lemma 7.3

(Anisotropic Sobolev inequalities in perforated domains)

• (i) For arbitrary $v \in H^1 (0,T; L^2 (\Omega _{\epsilon })) \cap L^2 (0,T; H^1 (\Omega _{\epsilon }))$ and $q_1$ and $r_1$ satisfying the conditions

  $$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\displaystyle 1}{\displaystyle r_1}+ \frac{\displaystyle N}{\displaystyle 2 q_1}=\frac{\displaystyle N}{\displaystyle 4} \\ r_1 \in [2, \infty ], \; q_1 \in \left[ 2, \frac{2 N}{N-2}\right] \; \; \; \text {for} \; N>2\\ \end{array}\right. } \end{aligned}$$

  (116)

  the following estimate holds

  $$\begin{aligned} \Vert v \Vert _{L^{r_1} (0, T; L^{q_1} (\Omega _{\epsilon }))} \le c \, \Vert v \Vert _{Q_{\epsilon } (T)} \end{aligned}$$

  (117)

  where c is a positive constant independent of $\epsilon$ and

  $$\begin{aligned} \Vert v \Vert ^{2}_{Q_{\epsilon } (T)} := \sup _{0 \le t \le T} \displaystyle \int _{\Omega _{\epsilon }} \vert v(t) \vert ^2 \, \mathrm{d}x+ \displaystyle \int _{0}^{T} \, \mathrm{d}t \displaystyle \int _{\Omega _{\epsilon }} \vert \nabla v(t) \vert ^2 \, \mathrm{d}x \end{aligned}$$

  (118)

• (ii) For arbitrary $v \in H^1 (0,T; L^2 (\Omega _{\epsilon })) \cap L^2 (0,T; H^1 (\Omega _{\epsilon }))$ and $q_2$ and $r_2$ satisfying the conditions

  $$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\displaystyle 1}{\displaystyle r_2}+ \frac{\displaystyle (N-1)}{\displaystyle 2 q_2}=\frac{\displaystyle N}{\displaystyle 4} \\ r_2 \in [2, \infty ], \; q_2 \in \left[ 2, \frac{2 (N-1)}{(N-2)}\right] \; \; \; \text {for} \; N \ge 3\\ \end{array}\right. } \end{aligned}$$

  (119)

  the following estimate holds

  $$\begin{aligned} \Vert v \Vert _{L^{r_2} (0, T; L^{q_2} (\Gamma _{\epsilon }))} \le c \, {\epsilon }^{-\frac{N}{2}-\frac{(1-N)}{q_2}} \, \Vert v \Vert _{Q_{\epsilon } (T)} \end{aligned}$$

  (120)

  where c is a positive constant independent of $\epsilon$ and the norm $\Vert v \Vert _{Q_{\epsilon } (T)}$ is defined as in (118).

Proof

(i) The extension Lemma 7.2 ensures the well definiteness of a linear continuous extension operator $P_{\epsilon }$ which satisfies (114) and (115). By the classical multiplicative Sobolev inequalities valid in $\Omega$ (see Ladyzenskaja et al. 1968 and Nittka 2014), we have that

$$\begin{aligned} \Vert P_{\epsilon } v \Vert _{L^{r_1} (0, T; L^{q_1} (\Omega ))} \le c_1 \, \Vert P_{\epsilon } v \Vert _{Q (T)} \end{aligned}$$

(121)

where $c_1 \ge 0$ depends only on $\Omega$, $r_1$, $q_1$, with $r_1$ and $q_1$ satisfying the conditions (116) and

$$\begin{aligned} \Vert P_{\epsilon } v \Vert ^{2}_{Q (T)} := \sup _{0 \le t \le T} \displaystyle \int _{\Omega } \vert P_{\epsilon } v(t) \vert ^2 \, \mathrm{d}x+ \displaystyle \int _{0}^{T} \, \mathrm{d}t \displaystyle \int _{\Omega } \vert \nabla (P_{\epsilon } v(t)) \vert ^2 \, \mathrm{d}x \end{aligned}$$

(122)

By using (114), (115) and (121), we conclude that

$$\begin{aligned} \begin{aligned} \Vert v \Vert _{L^{r_1} (0, T; L^{q_1} (\Omega _{\epsilon }))}&\le C' \, \Vert P_{\epsilon } v \Vert _{L^{r_1} (0, T; L^{q_1} (\Omega ))} \\&\le C' \, c_1 \, \Vert P_{\epsilon } v \Vert _{Q (T)} \le C' \, c_1 \, C \, \Vert v \Vert _{Q_{\epsilon } (T)} \end{aligned} \end{aligned}$$

(123)

where $c := C' \, c_1 \, C$ is independent of $\epsilon$.

(ii) Let us rewrite the anisotropic Sobolev inequality valid on $\partial \Omega$ (see Ladyzenskaja et al. 1968 and Nittka 2014):

$$\begin{aligned} \begin{aligned}&\bigg [ \displaystyle \int _{0}^{T} \, \mathrm{d}t \bigg [ \displaystyle \int _{\partial \Omega } \vert v(t) \vert ^{q_2} \, d\mathcal {H}^{N-1} \bigg ]^{\frac{r_2}{q_2}} \bigg ]^{\frac{1}{r_2}} \\&\le c_1 \; \bigg [ \sup _{0 \le t \le T} \displaystyle \int _{\Omega } \vert v(t) \vert ^2 \, \mathrm{d}y+ \displaystyle \int _{0}^{T} \, \mathrm{d}t \displaystyle \int _{\Omega } \vert \nabla v(t) \vert ^2 \, \mathrm{d}y \bigg ]^{1/2} \end{aligned} \end{aligned}$$

(124)

where $c_1 \ge 0$ depends only on $r_2$, $q_2$ and on local properties of the surface $\partial \Omega$ (which is assumed to be piecewise smooth) with $r_2$ and $q_2$ satisfying the conditions (119). By performing the change of variable $y=\frac{\displaystyle x}{\displaystyle \epsilon }$, it is easy to obtain the corresponding re-scaled estimates:

$$\begin{aligned}&\epsilon ^{\frac{(1-N)}{q_2}} \, \bigg [ \displaystyle \int _{0}^{T} \, \mathrm{d}t \bigg [ \displaystyle \int _{\Gamma _{\epsilon }} \vert v(t) \vert ^{q_2} \, d\mathcal {H}^{N-1} \bigg ]^{\frac{r_2}{q_2}} \bigg ]^{\frac{1}{r_2}} \nonumber \\&\quad \le c_1 \, \epsilon ^{-\frac{N}{2}} \, \bigg [ \sup _{0 \le t \le T} \displaystyle \int _{\Omega _{\epsilon }} \vert v(t) \vert ^2 \, \mathrm{d}x+ \epsilon ^{2} \displaystyle \int _{0}^{T} \, \mathrm{d}t \displaystyle \int _{\Omega _{\epsilon }} \vert \nabla v(t) \vert ^2 \, \mathrm{d}x \bigg ]^{1/2} \end{aligned}$$

(125)

$$\begin{aligned}&\bigg [ \displaystyle \int _{0}^{T} \, \mathrm{d}t \bigg [ \displaystyle \int _{\Gamma _{\epsilon }} \vert v(t) \vert ^{q_2} \, d\mathcal {H}^{N-1} \bigg ]^{\frac{r_2}{q_2}} \bigg ]^{\frac{1}{r_2}} \nonumber \\&\quad \le c \, \epsilon ^{-\frac{N}{2}-\frac{(1-N)}{q_2}} \, \bigg [ \sup _{0 \le t \le T} \displaystyle \int _{\Omega _{\epsilon }} \vert v(t) \vert ^2 \, \mathrm{d}x+ \displaystyle \int _{0}^{T} \, \mathrm{d}t \displaystyle \int _{\Omega _{\epsilon }} \vert \nabla v(t) \vert ^2 \, \mathrm{d}x \bigg ]^{1/2} \end{aligned}$$

(126)

where c is a positive constant independent of $\epsilon$. $\square$

Appendix B

Let us introduce some definitions and results on two-scale convergence from Allaire (1992), Allaire et al. (1996), Nguetseng (1989), slightly modified to allow for homogenization with a parameter (the time t) (Clark and Showalter 1999; Hornung 1992; Nandakumaran and Rajesh 2002).

Definition 7.1

A sequence of functions $v^{\epsilon }$ in $L^2 ([0,T] \times \Omega )$ two-scale converges to $v_0 \in L^2 ([0,T] \times \Omega \times Y)$ if

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \int _0^T \int _{\Omega } v^{\epsilon }(t,x) \, \phi \bigg ( t,x,\frac{x}{\epsilon } \bigg ) \,\mathrm{d}t \,\mathrm{d}x= \int _0^T \int _{\Omega } \int _{Y} v_0 (t,x,y) \, \phi (t,x,y) \,\mathrm{d}t \, \mathrm{d}x\, \mathrm{d}y \end{aligned}$$

(127)

for all $\phi \in C^1 ([0,T] \times \overline{\Omega }; C_{\#}^{\infty }(Y))$.

The notion of ‘two-scale convergence’ makes sense because of the next compactness theorem.

Theorem 7.1

If $v^{\epsilon }$ is a bounded sequence in $L^2 ([0,T] \times \Omega )$, then there exists a function $v_0 (t,x,y)$ in $L^2 ([0,T] \times \Omega \times Y)$ such that, up to a subsequence, $v^{\epsilon }$ two-scale converges to $v_0$.

The following theorem is useful in obtaining the limit of the product of two two-scale convergent sequences.

Theorem 7.2

Let $v^{\epsilon }$ be a sequence of functions in $L^2 ([0,T] \times \Omega )$ which two-scale converges to a limit $v_0 \in L^2 ([0,T] \times \Omega \times Y)$. Suppose, furthermore, that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \int _0^T \int _{\Omega } \vert v^{\epsilon }(t,x) \vert ^2 \, \mathrm{d}t \, \mathrm{d}x= \int _0^T \int _{\Omega } \int _{Y} \vert v_0 (t,x,y) \vert ^2 \, \mathrm{d}t \, \mathrm{d}x \, \mathrm{d}y \end{aligned}$$

(128)

Then, for any sequence $w^{\epsilon }$ in $L^2 ([0,T] \times \Omega )$ that two-scale converges to a limit $w_0 \in L^2 ([0,T] \times \Omega \times Y)$, we have

$$\begin{aligned} \begin{aligned} \lim _{\epsilon \rightarrow 0} \int _0^T \int _{\Omega } v^{\epsilon }(t,x) \,&w^{\epsilon }(t,x) \, \phi \bigg ( t,x,\frac{x}{\epsilon } \bigg ) \,\mathrm{d}t \,\mathrm{d}x \\ {}&= \int _0^T \int _{\Omega } \int _{Y} v_0 (t,x,y) \, w_0 (t,x,y) \, \phi (t,x,y) \,\mathrm{d}t \, \mathrm{d}x\, \mathrm{d}y \end{aligned} \end{aligned}$$

(129)

for all $\phi \in C^1 ([0,T] \times \overline{\Omega }; C_{\#}^{\infty }(Y))$.

The next theorems yield a characterization of the two-scale limit of the gradients of bounded sequences $v^{\epsilon }$. This result is crucial for applications to homogenization problems.

We identify $H^1(\Omega )=W^{1,2} (\Omega )$, where the Sobolev space $W^{1,p} (\Omega )$ is defined by

$$\begin{aligned} W^{1,p} (\Omega )=\bigg \{ v \vert v \in L^p(\Omega ), \frac{\partial v}{\partial x_i} \in L^p(\Omega ), i=1, \ldots , N \bigg \} \end{aligned}$$

and we denote by $H^1_{\#}(Y)$ the closure of $C^{\infty }_{\#}(Y)$ for the $H^1$-norm.

Theorem 7.3

Let $v^{\epsilon }$ be a bounded sequence in $L^2 (0,T; H^1 (\Omega ))$ that converges weakly to a limit v(t, x) in $L^2 (0,T; H^1 (\Omega ))$. Then, $v^{\epsilon }$ two-scale converges to v(t, x), and there exists a function $v_1 (t,x,y)$ in $L^2 ([0,T] \times \Omega ; H^1_{\#} (Y)/\mathbb {R})$ such that, up to a subsequence, $\nabla v^{\epsilon }$ two-scale converges to $\nabla _x v(t,x)+\nabla _y v_1 (t,x,y)$.

Theorem 7.4

Let $v^{\epsilon }$ and $\epsilon \nabla v^{\epsilon }$ be two bounded sequences in $L^2 ([0,T] \times \Omega )$. Then, there exists a function $v_1 (t,x,y)$ in $L^2 ([0,T] \times \Omega ; H^1_{\#} (Y)/\mathbb {R})$ such that, up to a subsequence, $v^{\epsilon }$ and $\epsilon \nabla v^{\epsilon }$ two-scale converge to $v_1 (t,x,y)$ and $\nabla _y v_1 (t,x,y)$, respectively.

The main result of two-scale convergence can be generalized to the case of sequences defined in $L^2 ([0,T] \times \Gamma _{\epsilon })$.

Theorem 7.5

Let $v^{\epsilon }$ be a sequence in $L^2 ([0,T] \times \Gamma _{\epsilon })$ such that

$$\begin{aligned} \epsilon \, \int _0^T \int _{\Gamma _{\epsilon }} \vert v^{\epsilon }(t,x) \vert ^2 \, \mathrm{d}t \, \mathrm{d}\sigma _{\epsilon }(x) \le C \end{aligned}$$

(130)

where C is a positive constant, independent of $\epsilon$. There exist a subsequence (still denoted by $\epsilon$) and a two-scale limit $v_0(t,x,y) \in L^2([0,T] \times \Omega ; L^2(\Gamma ))$ such that $v^{\epsilon }(t,x)$ two-scale converges to $v_0(t,x,y)$ in the sense that

$$\begin{aligned}&\lim _{\epsilon \rightarrow 0} \, \epsilon \, \displaystyle \int _0^T \int _{\Gamma _{\epsilon }} v^{\epsilon }(t,x) \, \phi \bigg (t,x, \frac{x}{\epsilon } \bigg ) \, \mathrm{d}t \, \mathrm{d}\sigma _{\epsilon }(x)\nonumber \\&\quad = \displaystyle \int _0^T \int _{\Omega } \int _{\Gamma } v_0(t,x,y) \, \phi (t,x,y) \, \mathrm{d}t \, \mathrm{d}x \, \mathrm{d}\sigma (y) \end{aligned}$$

(131)

for any function $\phi \in C^1 ([0, T] \times \overline{\Omega }; C_{\#}^{\infty }(Y))$.

The proof of Theorem 7.5 is very similar to the usual two-scale convergence theorem (Allaire 1992). It relies on the following lemma (Allaire et al. 1996):

Lemma 7.4

Let $B=C [\overline{\Omega }; C_{\#} (Y)]$ be the space of continuous functions $\phi (x, y)$ on $\overline{\Omega } \times Y$ which are Y-periodic in y. Then, B is a separable Banach space which is dense in $L^2 (\Omega ; L^2 (\Gamma ))$, and such that any function $\phi (x, y) \in B$ satisfies

$$\begin{aligned} \epsilon \, \displaystyle \int _{\Gamma _{\epsilon }} \bigg \vert \phi (x,\frac{x}{\epsilon } ) \bigg \vert ^2 \, \mathrm{d}\sigma _{\epsilon }(x) \le C \, \Vert \phi \Vert ^2_B, \end{aligned}$$

(132)

and

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \, \epsilon \, \int _{\Gamma _{\epsilon }} \bigg \vert \phi \bigg (x, \frac{x}{\epsilon } \bigg ) \bigg \vert ^2 \, \mathrm{d}\sigma _{\epsilon } (x) =\displaystyle \int _{\Omega } \int _{\Gamma } \vert \phi (x, y) \vert ^2 \, \mathrm{d}x \, \mathrm{d}\sigma (y). \end{aligned}$$

(133)