Compressible Navier-Stokes equations with heterogeneous pressure laws

TL;DR

This paper establishes the existence of global weak solutions for compressible Navier-Stokes equations with heterogeneous pressure laws depending on density, time, and space, using a novel regularization and fixed point approach.

Contribution

It introduces a new regularized fixed point method to construct solutions for Navier-Stokes equations with complex pressure dependencies, advancing the mathematical understanding of such models.

Findings

Proves existence of global weak solutions under broad conditions.

Develops a new regularization technique for pressure-dependent equations.

Provides a framework for future heat-conducting Navier-Stokes analysis.

Abstract

This paper concerns the existence of global weak solutions \`a la Leray for compressible Navier-Stokes equations with a pressure law that depends on the density and on time and space variables $t$ and $x$. The assumptions on the pressure contain only locally Lipschitz assumption with respect to the density variable and some hypothesis with respect to the extra time and space variables. It may be seen as a first step to consider heat-conducting Navier-Stokes equations with physical laws such as the truncated virial assumption. The paper focuses on the construction of approximate solutions through a new regularized and fixed point procedure and on the weak stability process taking advantage of the new method introduced by the two first authors with a careful study of an appropriate regularized quantity linked to the pressure.