Homogenization of the two-dimensional evolutionary compressible Navier–Stokes equations

Abstract

We consider the evolutionary compressible Navier–Stokes equations in a two-dimensional perforated domain, and show that in the subcritical case of very tiny holes, the density and velocity converge to a solution of the evolutionary compressible Navier–Stokes equations in the non-perforated domain.

Appendix A: Bogovskiĭ’s operator in 2D and improved pressure estimates

In this section, we give an inverse to the divergence in the two-dimensional perforated domain, and estimate its norm in any $L^q(D_\varepsilon )$. To the best of the authors’ knowledge, such estimates for two spatial dimensions are just known in the $L^2$-setting, see [24, Section 1.4]. Therefore, we give here an explicit proof, which might be of independent interest. As an application, we explain how to use it to prove Lemma 3.2.

Theorem A.1

Let $D\subset {\mathbb {R}}^2$ be a bounded domain with smooth boundary and $D_\varepsilon$ be defined as in (3). Then, there exists an operator ${\mathcal {B}}_\varepsilon$ such that for any $q\ge 1$,

$$\begin{aligned} {\mathcal {B}}_\varepsilon :L_0^q(D_\varepsilon )=\{f\in L^q(D_\varepsilon ):\int _{D_\varepsilon } f \, \textrm{d}x =0\}\rightarrow W_0^{1,q}(D_\varepsilon ;{\mathbb {R}}^2), \end{aligned}$$

and for any $f\in L_0^q(D_\varepsilon )$ we have

$$\begin{aligned} \textrm{div}\, {\mathcal {B}}_\varepsilon (f)=f \text { in } D_\varepsilon , \quad \Vert {\mathcal {B}}_\varepsilon \Vert _{W_0^{1,q}(D_\varepsilon )}^q \lesssim (1+C(\varepsilon ,q))\Vert f\Vert _{L^q(D_\varepsilon )}^q, \end{aligned}$$

where

$$\begin{aligned} C(\varepsilon ,q)=\varpi _\varepsilon ^{-2} a_\varepsilon ^{-q} {\left\{ \begin{array}{ll} |\log \varpi _\varepsilon |^{-q} |\varpi _\varepsilon ^{2-q}-1| &{} \text {if } q\ne 2,\\ |\log \varpi _\varepsilon |^{-1} &{} \text {if } q=2, \end{array}\right. } \end{aligned}$$

and $\varpi _\varepsilon$ is as in (19).

Proof

We follow the idea of [9], where $L^q$-estimates are given for the case of three spatial dimensions. Let $f\in L_0^q(D_\varepsilon )$. Then, there exists a function ${\textbf{u}}\in W_0^{1,q}(D)$ such that

$$\begin{aligned} \textrm{div}\,{\textbf{u}}=\tilde{f} \text { in } D, \quad \Vert {\textbf{u}}\Vert _{W_0^{1,q}(D)}\le C \Vert f\Vert _{L^q(D_\varepsilon )} \end{aligned}$$

for some constant $C>0$ independent of $\varepsilon$ (see [5, 13]). However, ${\textbf{u}}$ does not vanish on the holes in general. To overcome this, note that the domains $B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon )\setminus B_{a_\varepsilon }(z_i^\varepsilon )$ are uniform John domains (see, e.g., [27, Example 3.2.2]), so for each $i\in K_\varepsilon$, there exists a Bogovskiĭ operator ${\mathcal {B}}_{i,\varepsilon }$ satisfying

$$\begin{aligned}&{\mathcal {B}}_{i,\varepsilon }:L_0^q(B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon )\setminus B_{a_\varepsilon } (z_i^\varepsilon ))\rightarrow W_0^{1,q}(B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon )\setminus B_{a_\varepsilon }(z_i^\varepsilon )),\\&\textrm{div}\, B_{i,\varepsilon }(g)=g \text { in } B_{\varpi _\varepsilon a_\varepsilon } (z_i^\varepsilon )\setminus B_{a_\varepsilon }(z_i^\varepsilon ),\\ {}&\Vert {\mathcal {B}}_{i,\varepsilon }(g) \Vert _{L^q(B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon )\setminus B_{a_\varepsilon } (z_i^\varepsilon ))} \le C \Vert g\Vert _{L^q(B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon ) \setminus B_{a_\varepsilon }(z_i^\varepsilon ))} \end{aligned}$$

for some constant $C>0$ independent of $\varepsilon$ (see [8, Theorem 5.2]). Furthermore, we define $\mathfrak {y}_\varepsilon$ as in (12), and

$$\begin{aligned} \vartheta _\varepsilon (r)={\left\{ \begin{array}{ll} 1 &{} \text {if } 0\le r< \varpi _\varepsilon a_\varepsilon /2,\\ \frac{2}{\varpi _\varepsilon a_\varepsilon }(\varpi _\varepsilon a_\varepsilon -r) &{} \text {if } \varpi _\varepsilon a_\varepsilon /2\le r < \varpi _\varepsilon a_\varepsilon ,\\ 0 &{} \text {else}. \end{array}\right. } \end{aligned}$$

As before, set for $z_i^\varepsilon \in K_\varepsilon$ and $x\in D$ the functions $\eta _\varepsilon ^i(x)=\mathfrak {y}_\varepsilon (|x-z_i^\varepsilon |)$ and $\theta _\varepsilon ^i(x)=\vartheta _\varepsilon (|x-z_i^\varepsilon |)$, and define for ${\textbf{u}}\in W^{1,q}(B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon ))$ the operator $L_{i,\varepsilon }$ as

$$\begin{aligned} L_{i,\varepsilon } {\textbf{u}}(x) = \theta _\varepsilon ^i(x)\bigg ({\textbf{u}}(x) -\frac{1}{|B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon )|} \int _{B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon )} {\textbf{u}} \, \textrm{d}x \bigg ) + \eta _\varepsilon ^i(x) \frac{1}{|B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon )|} \int _{B_{\varpi _\varepsilon a_\varepsilon } (z_i^\varepsilon )} {\textbf{u}} \, \textrm{d}x. \end{aligned}$$

Note that this immediately implies $L_{i,\varepsilon }{\textbf{u}}=0$ on $\partial B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon )$ as well as $L_{i,\varepsilon }{\textbf{u}}={\textbf{u}}$ on $\partial B_{a_\varepsilon }(z_i^\varepsilon )$. Moreover, by Poincaré’s inequality,

$$\begin{aligned} \bigg \Vert {\textbf{u}}-\frac{1}{|B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon )|} \int _{B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon )} {\textbf{u}} \, \textrm{d}x \bigg \Vert _{L^q(B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon ))} \le C \varpi _\varepsilon a_\varepsilon \Vert \nabla {\textbf{u}}\Vert _{L^q(B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon ))} \end{aligned}$$

for some constant $C>0$ independent of $\varepsilon$. Hence, by the estimate (13) and Hölder’s inequality,

$$\begin{aligned}&\Vert \nabla L_{i,\varepsilon }{\textbf{u}}\Vert _{L^q(B_{\varpi _\varepsilon a_\varepsilon } (z_i^\varepsilon ))} \le \Vert \nabla \theta _\varepsilon ^i\Vert _{L^\infty (D)} \bigg \Vert {\textbf{u}}-\frac{1}{|B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon )|} \int _{B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon )} {\textbf{u}} \, \textrm{d}x \bigg \Vert _{L^q(B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon ))}\\&\qquad + \Vert \nabla {\textbf{u}}\Vert _{L^q(B_{\varpi _\varepsilon a_\varepsilon } (z_i^\varepsilon ))} + \Vert \nabla \eta _\varepsilon ^i\Vert _{L^q(B_{\varpi _\varepsilon a_\varepsilon } (z_i^\varepsilon ))} \frac{1}{|B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon )|} \int _{B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon )} |{\textbf{u}}| \, \textrm{d}x\\&\quad \lesssim \Vert \nabla {\textbf{u}}\Vert _{L^q(B_{\varpi _\varepsilon a_\varepsilon } (z_i^\varepsilon ))} + (\varpi _\varepsilon a_\varepsilon )^{-\frac{2}{q}} a_\varepsilon ^{\frac{2}{q}-1} \Vert {\textbf{u}}\Vert _{L^q(B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon ))} {\left\{ \begin{array}{ll} |\log \varpi _\varepsilon |^{-1} |\varpi _\varepsilon ^{2-q}-1|^\frac{1}{q} &{} \text {if } q\ne 2,\\ |\log \varpi _\varepsilon |^{-\frac{1}{2}} &{} \text {if } q=2, \end{array}\right. } \end{aligned}$$

yielding an operator $L_{i,\varepsilon }:W^{1,q}(B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon ))\rightarrow W_0^{1,q}(B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon ))$. Eventually, we define

$$\begin{aligned} {\mathcal {B}}_\varepsilon (f)={\textbf{u}} - \sum _{i\in K_\varepsilon } L_{i,\varepsilon }{\textbf{u}} -{\mathcal {B}}_{i,\varepsilon } \textrm{div}\, L_{i,\varepsilon } {\textbf{u}}. \end{aligned}$$

Note that this operator is well defined due to

$$\begin{aligned} \int _{B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon )} \textrm{div} \, L_{i,\varepsilon }{\textbf{u}} \, \textrm{d}x = \int _{\partial B_{\varpi _\varepsilon a_\varepsilon } (z_i^\varepsilon )} L_{i,\varepsilon }{\textbf{u}} \cdot {\textbf{n}} \, \textrm{d}\sigma = 0 \end{aligned}$$

since $L_{i,\varepsilon }{\textbf{u}}=0$ on $\partial B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon )$. Furthermore, for any $x\in B_{a_\varepsilon }(z_i^\varepsilon )$,

$$\begin{aligned} \textrm{div}\, L_{i,\varepsilon }{\textbf{u}}(x)=\textrm{div}\, {\textbf{u}}(x) =\tilde{f}(x)=0 \end{aligned}$$

by $L_{i,\varepsilon }{\textbf{u}}={\textbf{u}}$ in $B_{a_\varepsilon }(z_i^\varepsilon )$ and $\tilde{f}(x)=0$ on $D\setminus D_\varepsilon$. Hence,

$$\begin{aligned} \int _{B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon )\setminus B_{a_\varepsilon }(z_i^\varepsilon )} \textrm{div}\, L_{i,\varepsilon }{\textbf{u}}\, \textrm{d}x = 0 \end{aligned}$$

as wished. Moreover, this leads for any $x\in B_{a_\varepsilon }(z_i^\varepsilon )$ to

$$\begin{aligned} {\mathcal {B}}_\varepsilon (f)(x)={\textbf{u}}(x)-L_{i,\varepsilon }{\textbf{u}}(x)=0, \end{aligned}$$

so indeed ${\mathcal {B}}_\varepsilon (f)=0$ on $D\setminus D_\varepsilon$. Seeing finally that the holes $B_{\varpi _\varepsilon a_\varepsilon }(z_i^\varepsilon )$ are disjoint, we sum up the estimates obtained to finish the proof of the theorem. $\square$

With the help of the operator ${\mathcal {B}}_\varepsilon$, we can show Lemma 3.2. Recalling $a_\varepsilon =\exp (-\varepsilon ^{-\alpha })$ for some $\alpha >2$ and $\varpi _\varepsilon = \varepsilon ^{1+\delta }a_\varepsilon ^{-1}$ from (19), we have for any $1\le q<2$ the bound

$$\begin{aligned} 1+C(\varepsilon ,q)&\lesssim 1 + \varpi _\varepsilon ^{-2} a_\varepsilon ^{-q} | \log \varpi _\varepsilon |^{-q} \varpi _\varepsilon ^{2-q} = 1 +(a_\varepsilon \varpi _\varepsilon | \log \varpi _\varepsilon |)^{-q} \\&\lesssim 1+\varepsilon ^{q(\alpha -1-\delta )} \lesssim 1, \end{aligned}$$

which is uniform as long as $\delta \le \alpha -1$. Similarly, for $q=2$, we have

$$\begin{aligned} 1+C(\varepsilon ,2)\lesssim 1+(\varpi _\varepsilon a_\varepsilon )^{-2} |\log \varpi _\varepsilon |^{-1} \lesssim 1+\varepsilon ^{\alpha -2(1+\delta )}\lesssim 1 \end{aligned}$$

as long as $\delta \le \frac{\alpha -2}{2}$. The idea is now to test the momentum equation (7) by the function

$$\begin{aligned} \varphi (t,x)=\xi (t){\mathcal {B}}_\varepsilon \bigg [\varrho _\varepsilon ^\theta -\frac{1}{|D_\varepsilon |} \int _{D_\varepsilon } \varrho _\varepsilon ^\theta \bigg ] \end{aligned}$$

for $\theta <\gamma -1$ and some $\xi \in C_c^\infty ([0,T))$. Note that the function $\varphi$ is not regular enough in the time variable to use it as test function, however, one can overcome this by using a time-regularization argument (see [10, Section 2.2.5] for details). The proof of the improved integrability of the density now follows the same lines as [6, Appendix B] (see also [27, Section 4.2.2]).