Existence and stability of steady noncharacteristic solutions on a finite interval of full compressible Navier-Stokes equations

TL;DR

This paper investigates the existence, uniqueness, and stability of steady solutions to the full 1D compressible Navier-Stokes equations in a shock tube setting, highlighting conditions for stability and demonstrating nonuniqueness in certain artificial models.

Contribution

It establishes the existence of steady noncharacteristic solutions for the full compressible Navier-Stokes equations with arbitrary data and explores their stability and nonuniqueness in specific cases.

Findings

Existence of steady solutions for full polytropic gas dynamics.

Numerical evidence for uniqueness and stability of these solutions.

Example of nonuniqueness in artificial equations of state.

Abstract

We treat the 1D shock tube problem, establishing existence of steady solutions of full (nonisentropic) polytropic gas dynamics with arbitrary noncharacteristic data. We present also numerical experiments indicating uniqueness and time-asymptotic stability of such solutions. At the same time, we give an example of an (artificial) equation of state possessing a convex entropy for which there holds nonuniqueness of solutions. This is associated with instability and Hopf bifurcation to time-periodic solutions. In a second part, we study general systems of viscous conservation laws and we state results on existence, stability and spectral stability of steady states in restricted cases.