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

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.