Two-Dimensional Vortex Sheets for the Nonisentropic Euler Equations: Nonlinear Stability

TL;DR

This paper proves the short-time existence and nonlinear stability of vortex sheets in two-dimensional nonisentropic Euler equations, using energy estimates and Nash--Moser iteration based on linear stability analysis.

Contribution

It establishes the nonlinear stability of vortex sheets for nonisentropic Euler equations, extending previous linear stability results to a nonlinear setting.

Findings

Short-time existence of vortex sheets proven

Nonlinear stability established in Sobolev spaces

Method based on tame energy estimates and Nash--Moser scheme

Abstract

We show the short-time existence and nonlinear stability of vortex sheets for the nonisentropic compressible Euler equations in two spatial dimensions, based on the weakly linear stability result of Morando--Trebeschi (2008) [20]. The missing normal derivatives are compensated through the equations of the linearized vorticity and entropy when deriving higher-order energy estimates. The proof of the resolution for this nonlinear problem follows from certain \emph{a priori} tame estimates on the effective linear problem {in the usual Sobolev spaces} and a suitable Nash--Moser iteration scheme.