Global Existence of Weak Solutions for Compresssible Navier--Stokes Equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor

TL;DR

This paper establishes the global existence of weak solutions for the compressible Navier--Stokes equations with generalized, thermodynamically unstable pressure laws and anisotropic viscous stress tensors, expanding the mathematical framework for complex physical phenomena.

Contribution

It introduces new analytical techniques to handle unstable pressure laws and anisotropic stresses, broadening the applicability of the theory to physical models.

Findings

Proved global existence of weak solutions under new conditions.

Developed novel compactness and regularity estimates for density.

Extended the theory to include physically relevant anisotropic stresses.

Abstract

We prove global existence of appropriate weak solutions for the compressible Navier--Stokes equations for more general stress tensor than those covered by P.-L. Lions and E. Feireisl's theory. More precisely we focus on more general pressure laws which are not thermodynamically stable; we are also able to handle some anisotropy in the viscous stress tensor. To give answers to these two longstanding problems, we revisit the classical compactness theory on the density by obtaining precise quantitative regularity estimates: This requires a more precise analysis of the structure of the equations combined to a novel approach to the compactness of the continuity equation. These two cases open the theory to important physical applications, for instance to describe solar events (virial pressure law), geophysical flows (eddy viscosity) or biological situations (anisotropy).