On the evolution equation of compressible vortex sheets

A. Morando, P. Secchi and P. Trebeschi

Mathematische Nachrichten, 2020, vol. 293, issue 5, 945-969

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 Mach number M satisfies M 2 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.

Date: 2020
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https://doi.org/10.1002/mana.201800162

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