On the stability of multi-dimensional rarefaction waves I: the energy estimates

TL;DR

This paper establishes the non-linear stability of multi-dimensional rarefaction waves in gas dynamics using novel energy estimates that avoid loss of derivatives, providing foundational results for understanding discontinuities in fluid flows.

Contribution

It introduces new energy estimates without loss of derivatives to prove stability of multi-dimensional rarefaction waves in Euler equations.

Findings

Proved non-linear stability of multi-dimensional rarefaction waves.

Developed energy estimates that do not lose derivatives.

Provided geometric descriptions of wave fronts.

Abstract

We study the resolution of discontinuous singularities in gas dynamics via rarefaction waves. The mechanism is well-understood in the one dimensional case. We will prove the non-nonlinear stability of the Riemann problem for multi-dimensional isentropic Euler equations in the regime of rarefaction waves. The proof relies on the new energy estimates \emph{without loss of derivatives}. We also give a detailed geometric description of the rarefaction wave fronts. This is the first paper in the series which provides the \emph{a priori} energy bounds.