Global weak solutions of PDEs for compressible media: A compactness criterion to cover new physical situations

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 instancemodel flows in a compressible tissue in biology or in a compressible porousmedia in petroleumengineering. 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.