Vanishing viscosity limit of the compressible Navier-Stokes equations with general pressure law

TL;DR

This paper proves that solutions of the one-dimensional compressible Navier-Stokes equations with general pressure laws converge to solutions of the Euler equations as viscosity vanishes, using entropy solutions and measure-valued analysis.

Contribution

It introduces a novel approach to handle unbounded densities and general pressure laws in the vanishing viscosity limit for compressible flows.

Findings

Convergence of Navier-Stokes to Euler equations established.

Construction of controlled fundamental solutions for entropy equations.

Reduction of measure-valued solutions to Dirac masses under Tartar's relation.

Abstract

We prove the convergence of the vanishing viscosity limit of the one-dimensional, isentropic, compressible Navier-Stokes equations to the isentropic Euler equations in the case of a general pressure law. Our strategy relies on the construction of fundamental solutions to the entropy equation that remain controlled for unbounded densities, and employs an improved reduction framework to show that measure-valued solutions constrained by the Tartar commutation relation (but with possibly unbounded support) reduce to a Dirac mass. As the Navier-Stokes equations do not admit an invariant region, we work in the finite-energy setting, where a detailed understanding of the high density regime is crucial.