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On the evolution equation of compressible vortex sheets
Author(s) -
Morando A.,
Secchi P.,
Trebeschi P.
Publication year - 2020
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800162
Subject(s) - mathematics , inviscid flow , sobolev space , euler equations , mathematical analysis , vortex , vortex sheet , compressibility , compressible flow , mach number , discontinuity (linguistics) , classical mechanics , vorticity , physics , mechanics
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 , the symbol is elliptic and the problem is ill‐posed. On the contrary, if M > 2then 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.