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

TL;DR

This paper proves the structural stability of complex wave configurations in 2D compressible Euler equations, including rarefaction waves and discontinuities, using energy estimates and nonlinear superposition analysis.

Contribution

It introduces a novel method to analyze the nonlinear superpositions of waves in gas dynamics without loss of derivatives.

Findings

Established stability of mixed wave configurations in 2D gas dynamics.

Derived energy estimates for acoustic and vorticity waves within rarefaction regions.

Analyzed nonlinear superpositions of shock, rarefaction, and vortex sheet waves.

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 $\Sigma_0=\{(t, x_1, x_2): t=0, (x_1, x_2)\in \mathbb{R}\times\mathbb{R}/2\pi\mathbb{Z}\}$ with discontinuities at $\mathbf{S}_*:=\{(t,x_{1},x_{2})\mid t=0,\ x_{1}=0, x_2\in\mathbb{R}/2\pi\mathbb{Z} \}$.