Inviscid limit of the compressible Navier-Stokes equations for asymptotically isothermal gases

TL;DR

This paper proves the convergence of solutions from the compressible Navier-Stokes equations to the Euler equations for asymptotically isothermal gases using compensated compactness and entropy methods.

Contribution

It introduces a novel approach with hyperbolic representation formulas to rigorously justify the inviscid limit for asymptotically isothermal gases.

Findings

Established existence of vanishing viscosity solutions

Proved convergence to Euler equations

Developed new entropy representation formulas

Abstract

We prove the existence of relative finite-energy vanishing viscosity solutions of the one-dimensional, isentropic Euler equations under the assumption of an asymptotically isothermal pressure law, that is, $p(\rho)/\rho = O(1)$ in the limit $\rho \to \infty$. This solution is obtained as the vanishing viscosity limit of classical solutions of the one-dimensional, isentropic, compressible Navier--Stokes equations. Our approach relies on the method of compensated compactness to pass to the limit rigorously in the nonlinear terms. Key to our strategy is the derivation of hyperbolic representation formulas for the entropy kernel and related quantities; among others, a special entropy pair used to obtain higher uniform integrability estimates on the approximate solutions. Intricate bounding procedures relying on these representation formulas then yield the required compactness of the entropy dissipation measures. In turn, we prove that the Young measure generated by the classical solutions of the Navier--Stokes equations reduces to a Dirac mass, from which we deduce the required convergence to a solution of the Euler equations.