On the evolution equation of compressible vortex sheets

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 satisfies < √ 2, the symbol is elliptic and the problem is ill-posed. On the contrary, if > √ 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. K E Y W O R D S compressible Euler equations, contact discontinuities, linear stability, loss of derivatives, vortex sheet, weak stability M S C ( 2 0 1 0 ) 35L50, 35Q35, 76E17, 76N10 √ 2 and are violently unstable when < √ 2, while planar vortex sheets are always violently unstable in three space dimensions. This kind of instabilities is the analogue of the Kelvin-Helmholtz instability for incompressible fluids.Artola and Majda [1] studied certain instabilities of two-dimensional supersonic vortex sheets by analyzing the interaction with highly oscillatory waves through geometric optics. A rigorous mathematical theory on nonlinear stability and local-in-time existence of two-dimensional supersonic vortex sheets was first established by 9] based on their linear stability results in [7] and a Nash-Moser iteration scheme.Characteristic discontinuities, especially vortex sheets, arise in a broad range of physical problems in fluid mechanics, oceanography, aerodynamics, plasma physics, astrophysics, and elastodynamics. The linear results in [7] have been generalized to cover the two-dimensional nonisentropic flows [15], the three-dimensional compressible steady flows [25,27], and the two-dimensional two-phase flows [18]. Recently, the methodology in [7] has been developed to deal with several constant Mathematische Nachrichten. 2020;293:945-969.

Nonlinear stability and existence of vortex sheets for inviscid liquid-gas two-phase flow

“…Later on, the result on the linear stability for the isentropic Euler equations was extended to the non-isentropic Euler equations in [27] (see also [28] for a very recent preprint on nonlinear stability). Recently, Morando-Secchi-Trebeschi in [26] provided an alternative way to prove the linear stability of compressible vortex sheets in two space dimensions using the evolution equation for the discontinuity front of the vortex sheet. For the MHD equations, the linear and nonlinear stability of compressible current-vortex sheets was proved in [2,37,41].…”

Anisotropic regularity of linearized compressible vortex sheets

“…For the reader's convenience we recall the main steps of the derivation of the evolution equation for the discontinuity front of the vortex sheet with constant coefficients, obtained in [10].…”

“…A rigorous mathematical theory on nonlinear stability and localin-time existence of two-dimensional supersonic vortex sheets was first established by Coulombel-Secchi [4,5] based on their linear stability results in [3] and a Nash-Moser iteration scheme. We refer the reader to [10] for more references.…”

“…In this paper 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 derived an evolution pseudo-differential equation for the discontinuity front of the vortex sheet. In agreement with the classical stability analysis [6,9], if the Mach number M < √ 2 the symbol is elliptic and the problem is ill-posed.…”

“…The aim of the present paper is to improve the result of [10] by showing the existence of the solution in a function space with some additional weighted anisotropic regularity in the frequency space. The main result of the paper is given by Theorem 3.2 in Section 3.…”