Anisotropic regularity of linearized compressible vortex sheets

TL;DR

This paper investigates the regularity properties of supersonic vortex sheets in compressible fluids, improving existing results by establishing solutions with enhanced anisotropic regularity in frequency space.

Contribution

It extends previous work by demonstrating the existence of solutions with weighted anisotropic regularity, refining the understanding of vortex sheet stability.

Findings

Existence of solutions with anisotropic regularity

Improved understanding of vortex sheet stability

Enhanced function space regularity results

Abstract

We are concerned with supersonic vortex sheets for the Euler equations of compressible inviscid fluids in two space dimensions. For the problem with constant coefficients, in [10] the authors have derived a pseudo-differential equation which describes the time evolution of the discontinuity front of the vortex sheet. In agreement with the classical stability analysis, the problem is weakly stable if $|[v\cdot\tau]|>2\sqrt{2}\,c$, and the well-posedness was obtained in standard weighted Sobolev spaces. The aim of the present paper is to improve the result of [10], by showing the existence of the solution in function spaces with some additional weighted anisotropic regularity in the frequency space.