On the stability of multi-dimensional rarefaction waves II: existence of solutions and applications to Riemann problem

TL;DR

This paper proves the stability of multi-dimensional rarefaction waves in gas dynamics, showing that smooth perturbations can be connected to rarefaction waves and that the Riemann problem remains stable under such conditions.

Contribution

It establishes the existence and stability of solutions near rarefaction waves in multi-dimensional gas dynamics, extending previous results with new energy estimates and applications.

Findings

Demonstrates stability of multi-dimensional rarefaction waves.

Connects smooth perturbations to rarefaction waves.

Shows structural stability of the Riemann problem in this regime.

Abstract

This is the second paper in a series studying the nonlinear stability of rarefaction waves in multi-dimensional gas dynamics. We construct initial data near singularities in the rarefaction wave region and, combined with the a priori energy estimates from the first paper, demonstrate that any smooth perturbation of constant states on one side of the diaphragm in a shock tube can be connected to a centered rarefaction wave. We apply this analysis to study multi-dimensional perturbations of the classical Riemann problem for isentropic Euler equations. We show that the Riemann problem is structurally stable in the regime of two families of rarefaction waves.