Multi-Dimensional Structural Stability of Mixed Riemann Configurations Containing Centered Rarefaction Waves and Surfaces of Discontinuities of Gas Dynamics

Abstract

For 2D compressible Euler equations of isentropic gas, we prove the structural stability of mixed Riemann configurations containing centered rarefaction waves and surfaces of discontinuities (such as shock waves or vortex sheets), by deriving simultaneous energy estimates for acoustic and vorticity waves within the rarefaction wave region \emph{without loss of derivatives} and examinations of the nonlinear superpositions of rarefaction waves with other waves such as shock waves or vortex sheets. The nonlinear superpositions of \emph{shock wave-rarefaction wave} and \emph{rarefaction wave-vortex sheet-rarefaction wave} are achieved by reducing the problems in corner regions to the Cauchy problems with the data prescribed on the plane $\hat{I}{\pounds }_0=\{(t,x_1,x_2):t=0,(x_1,x_2)\in \mathbb{R}\times \mathbb{R}/2\ddot{I}€\mathbb{Z}\}$ with discontinuities at ${\mathbf{S}}_{\ast }:=\{(t,x_1,x_2)\mid t=0,x_1=0,x_2\in \mathbb{R}/2\ddot{I}€\mathbb{Z}\}$.

Revisions include: updates to the title, abstract, and introduction; improved presentation of selected formulas and figures