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On the stability of multi-dimensional rarefaction waves with vorticity

Jin Jia, Tao Luo

Abstract

We study the stability of centered rarefaction waves for 2D isentropic compressible Euler equations for polytropic gas. The proof relies on the energy estimates on acoustical waves and vorticity wave simultaneously \emph{without loss of derivatives} in rarefaction wave region via reformulating the Euler equation as a wave-transport system introduced by Luk-Speck in \cite{LukSpeck2D}, which removes the irrotational assumption of Luo-Yu \cite{Luo-YuRare1, Luo-YuRare2}. As applications, we show that the local existence of piecewise continuous solutions to the Riemann problem in the perturbative regime of \emph{shock wave-rarefaction wave} and \emph{rarefaction wave-vortex sheet-rarefaction wave}.

On the stability of multi-dimensional rarefaction waves with vorticity

Abstract

We study the stability of centered rarefaction waves for 2D isentropic compressible Euler equations for polytropic gas. The proof relies on the energy estimates on acoustical waves and vorticity wave simultaneously \emph{without loss of derivatives} in rarefaction wave region via reformulating the Euler equation as a wave-transport system introduced by Luk-Speck in \cite{LukSpeck2D}, which removes the irrotational assumption of Luo-Yu \cite{Luo-YuRare1, Luo-YuRare2}. As applications, we show that the local existence of piecewise continuous solutions to the Riemann problem in the perturbative regime of \emph{shock wave-rarefaction wave} and \emph{rarefaction wave-vortex sheet-rarefaction wave}.
Paper Structure (70 sections, 28 theorems, 234 equations, 1 figure)

This paper contains 70 sections, 28 theorems, 234 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of Riemann problem for 2-D isentropic Euler system with 1-D symmetry
  3. The background solution
  4. The Riemann problem with configuration shock wave-rarefaction wave
  5. The Riemann problem with configuration rarefaction wave-vortex sheet-rarefaction wave
  6. Assumptions on the initial data
  7. The configuration of rarefaction wave-rarefaction wave
  8. The configuration of shock wave-rarefaction wave
  9. The configuration of rarefaction wave-vortex sheet-rarefaction wave
  10. Previous Works
  11. The acoustical geometry of Euler equations
  12. Riemann invariants, specific vorticity and the wave-transport equations
  13. The acoustical geometry and the construction of the first null frame
  14. Euler equations in the diagonal form
  15. The geometry of the second null frame
  16. ...and 55 more sections

Key Result

Proposition 2.3

Given $\mathring{U}_{r}$ and sufficiently small $\varepsilon$. For any smooth $U_{r}|_{t=0}$ defined on the right $x_{1}\geq 0$ up to the boundary with and any $\delta\in (0, \frac{1}{2})$, there exists $C^N$ data for rarefaction waves.

Figures (1)

  • Figure 1: Particle paths (integral curves of the material vector field $B$) transverse to rarefaction fronts away from vacuum.

Theorems & Definitions (58)

  • Remark 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5: Einstein summation convention
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • proof
  • Remark 2.1
  • ...and 48 more