Vanishing conductivity limit for the 1D compressible Navier-Stokes system

TL;DR

This paper analyzes the behavior of solutions to the 1D compressible Navier-Stokes equations with weak thermal conductivity, establishing bounds that remain stable as conductivity approaches zero.

Contribution

It provides a new proof of stability and rigorously demonstrates the zero-conductivity limit within the 'à la Hoff' solution framework.

Findings

Bounds that do not blow up as conductivity tends to zero

A new stability proof for the no-conductivity case

Convergence to the Navier-Stokes system without conduction

Abstract

The present article studies solutions to the compressible Navier-Stokes equations for ideal gases in one dimension when thermal conductivity is present but very weak, while viscosity is positive and constant. The main novelty is the establishment of bounds that do not explode when the conductivity coefficient approaches zero. The conductivity coefficient is assumed to be constant and the framework is that of ''{\`a} la Hoff'' solutions. More precisely, the velocity is initially assumed to be regular, while the density and temperature are only in L^infini and far from zero. A new proof of a stability result for cases without conductivity is given. Then, the proof of the zero-conductivity limit to the Navier-Stokes system without conduction is established in the ''{\`a} la Hoff'' framework.