We are concerned with supersonic vortex sheets for the Euler equations of nonisentropic 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 wave-type pseudo-differential equation of order two, whose symbol exhibits some poles as well as the source term. A careful analysis close to those poles is employed in order to maintain the microlocal structure of the symbol, in such a way as to remove the poles from both the operator and the source term. In the supersonic case, the problem is weakly stable, and we are able to derive an a priori energy estimate for the solution of the evolution equation for the front. Based on that, we prove the existence of the solution in weighted Sobolev spaces. This result extends the one in [15] for the isentropic case.
Two-dimensional nonisentropic compressible vortex sheets
Morando, Alessandro;Secchi, Paolo;Trebeschi, Paola;
2025-01-01
Abstract
We are concerned with supersonic vortex sheets for the Euler equations of nonisentropic 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 wave-type pseudo-differential equation of order two, whose symbol exhibits some poles as well as the source term. A careful analysis close to those poles is employed in order to maintain the microlocal structure of the symbol, in such a way as to remove the poles from both the operator and the source term. In the supersonic case, the problem is weakly stable, and we are able to derive an a priori energy estimate for the solution of the evolution equation for the front. Based on that, we prove the existence of the solution in weighted Sobolev spaces. This result extends the one in [15] for the isentropic case.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
