Global Weak Solutions of PDEs for Compressible Media: A Compactness Criterion to Cover New Physical Situations

Abstract

This short paper is an introduction of the memoir recently written by the two authors (see Bresch and Jabin, Global existence of weak solutions for compressible Navier–Stokes equations: thermodynamically unstable pressure and anisotropic viscous stress tensor, arXiv:1507.04629, 2015, submitted) which concerns the resolution of two longstanding problems: Global existence of weak solutions for compressible Navier–Stokes equations with thermodynamically unstable pressure and with anisotropic stress tensor. We focus here on a Stokes-like system which can for instance model flows in a compressible tissue in biology or in a compressible porous media in petroleum engineering. This allows to explain, on a simpler but still relevant and important system, the tools recently introduced by the authors and to discuss the important results that have been obtained on the compressible Navier–Stokes equations. It is finally a real pleasure to dedicate this paper to G. Métivier for his 65’s Birthday.

D. Bresch is partially supported by the ANR- 13-BS01-0003-01 project DYFICOLTI.

P.–E. Jabin is partially supported by NSF Grant 1312142 and by NSF Grant RNMS (Ki-Net) 1107444

Appendix

In this appendix, let us give different results which are used in the paper. The interested reader is referred to [4] for details and proofs but also [14]. These concern Maximal functions, Square functions and translation of operators. First we remind the well known inequality

$$\displaystyle{ \vert \varPhi (x) -\varPhi (y)\vert \leq C\,\vert x - y\vert \,(M\vert \nabla \varPhi \vert (x) + M\vert \nabla \varPhi \vert (y)), }$$

(36)

where M is the localized maximal operator

$$\displaystyle{ M\,f(x) =\sup _{r\leq 1} \frac{1} {\vert B(0,r)\vert }\,\int _{B(0,r)}f(x + z)\,dz. }$$

(37)

Let us mention several mathematical properties that may be proved, see [4]. First one has

Lemma A.1.

There exists C > 0 s.t. for any $u \in W^{1,1}(\mathbb{T}^{d})$ , one has

$$\displaystyle{\vert u(x) - u(y)\vert \leq C\,\vert x - y\vert \,(D_{\vert x-y\vert }u(x) + D_{\vert x-y\vert }u(y)),}$$

where we denote

$$\displaystyle{D_{h}u(x) = \frac{1} {h}\,\int _{\vert z\vert \leq h}\frac{\vert \nabla u(x + z)\vert } {\vert z\vert ^{d-1}} \,dz.}$$

Note that this result implies the estimate (36) as

Lemma A.2.

There exists C > 0, for any $u \in W^{1,p}(\mathbb{T}^{d})$ with p ≥ 1

$$\displaystyle{D_{h}\,u(x) \leq C\,M\vert \nabla u\vert (x).}$$

The key improvement in using D _h is that small translations of the operator D _h are actually easy to control

Lemma A.3.

Let $u \in H^{1}(\mathbb{T}^{d})$ then have the following estimates

$$\displaystyle{ \int _{h_{0}}^{1}\int _{ \mathbb{T}^{d}}\overline{K}_{h}(z)\,\|D_{\vert z\vert }\,u(.) - D_{\vert z\vert }\,u(. + z)\|_{L^{2}}\,dz\,\frac{dh} {h} \leq C\,\vert \log h_{0}\vert ^{1/2}\,\|u\|_{ H^{1}}. }$$

(38)

This lemma is critical and explain why we propagate a quantity integrated with respect to h with a weight dh∕h namely with the Kernel $\mathcal{K}_{h_{0}}$. The full proof is rather classical and can be found in [4] for any L ^p space but we give a brief sketch here (which is simpler as Lemma A.3 is L ² based and we can use Fourier transform).

Proof (Sketch of the proof of Lemma  A.3 .).

Note that we can write

$$\displaystyle{D_{h}\,u(x) = L_{h} \star \nabla u,\quad L(x) = \frac{1_{\vert x\vert \leq 1}} {\vert x\vert ^{d-1}},\quad L_{h}(z) = h^{-d}\,L(z/h),}$$

where L _h is hence a usual convolution operator and L ∈ W ^s, 1 for any s < 1. Now

$$\displaystyle\begin{array}{rcl} & & \int _{h_{0}}^{1}\int _{ \mathbb{T}^{d}}\overline{K}_{h}(z)\,\|D_{\vert z\vert }\,u(.) - D_{\vert z\vert }\,u(. + z)\|_{L^{2}}\,dz\,\frac{dh} {h} {}\\ & & \qquad \leq C\,\int _{S^{d-1}}\int _{0}^{1}\|L_{ r} \star \nabla u(.) - L_{r} \star \nabla u(. + r\,\omega )\|_{L^{2}}\, \frac{dr} {r + h_{0}}\,d\omega {}\\ & & \qquad \leq C\,\vert \log h_{0}\vert ^{1/2}\,\left (\int _{ S^{d-1}}\int _{0}^{1}\|L_{ r} \star \nabla u(.) - L_{r} \star \nabla u(. + r\,\omega )\|_{L^{2}}^{2}\, \frac{dr} {r + h_{0}}\,d\omega \right )^{1/2}. {}\\ \end{array}$$

For any ω ∈ S ^d−1, define L _r ^ω = L _r (. ) − L _r (. + r ω). Calculate by Fourier transform

$$\displaystyle{\int _{0}^{1}\|L_{ r} \star \nabla u(.) - L_{r} \star \nabla u(. + r\,\omega )\|_{L^{2}}^{2}\, \frac{dr} {r + h_{0}} =\int _{ 0}^{1}\sum _{ \xi \in \mathbb{T}^{d}}\vert \hat{L}_{r}^{\omega }\vert ^{2}(\xi )\,\vert \xi \vert ^{2}\vert \hat{u}\vert ^{2}(\xi )\, \frac{dr} {r + h_{0}}.}$$

But $\hat{L}_{r}^{\omega } = (1 - e^{ir\,\xi \cdot \omega })\,\hat{L}(r\,\xi )$ and furthermore $\vert \hat{L}(r\,\xi )\vert \leq C\,(1 + \vert r\,\xi \vert )^{-s}$ for some s > 0 since L ∈ W ^s, 1. Therefore

$$\displaystyle{\int _{0}^{1}\vert \hat{L}_{ r}^{\omega }\vert ^{2}(\xi )\, \frac{dr} {r + h_{0}} \leq C,}$$

for some constant C independent of ξ, ω and h ₀. This is of course the famous square function calculation and lets us conclude.