On the evolution equation of compressible vortex sheets

TL;DR

This paper derives an evolution equation for supersonic vortex sheets in compressible fluids, analyzing stability conditions and establishing well-posedness in weighted Sobolev spaces.

Contribution

It introduces a pseudo-differential evolution equation for vortex sheet discontinuities and characterizes stability regimes based on velocity jumps.

Findings

Elliptic symbol when velocity jump is below threshold, indicating ill-posedness.

Weak stability and energy estimates when velocity jump exceeds threshold.

Proof of well-posedness in weighted Sobolev spaces for the problem.

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 we derive an evolution equation for the discontinuity front of the vortex sheet. This is a pseudo-differential equation of order two. In agreement with the classical stability analysis, if the jump of the tangential component of the velocity satisfies $|[v\cdot\tau]|<2\sqrt{2}\,c$ (here $c$ denotes the sound speed) the symbol is elliptic and the problem is ill-posed. On the contrary, if $|[v\cdot\tau]|>2\sqrt{2}\,c$, then the problem is weakly stable, and we are able to derive a wave-type a priori energy estimate for the solution, with no loss of regularity with respect to the data. Then we prove the well-posedness of the problem, by showing the existence of the solution in weighted Sobolev spaces.