Nonlinear compressible vortex sheets in two space dimensions

Abstract

Aʙʀ. -We consider supersonic compressible vortex sheets for the isentropic Euler equations of gas dynamics in two space dimensions. The problem is a free boundary nonlinear hyperbolic problem with two main difficulties: the free boundary is characteristic, and the so-called Lopatinskii condition holds only in a weak sense, which yields losses of derivatives. Nevertheless, we prove the local existence of such piecewise smooth solutions to the Euler equations. Since the a priori estimates for the linearized equations exhibit a loss of regularity, our existence result is proved by using a suitable modification of the Nash-Moser iteration scheme. We also show how a similar analysis yields the existence of weakly stable shock waves in isentropic gas dynamics, and the existence of weakly stable liquid/vapor phase transitions.R. -Nous construisons des nappes de tourbillon supersoniques pour les équations d'Euler compressibles isentropiques en deux dimensions d'espace. Il s'agit d'un problème non-linéaire hyperbolique à frontière libre présentant deux difficultés principales : la frontière libre est caractéristique et la condition dite de Lopatinskii n'est satisfaite que dans un sens faible, ce qui induit des estimations à perte. Néanmoins nous montrons l'existence de telles solutions régulières par morceaux des équations d'Euler en utilisant un schéma itératif de type Nash-Moser palliant les pertes de régularité. Notre analyse s'étend au cas de discontinuités non-caractéristiques et faiblement stables comme certaines ondes de choc pour les équations d'Euler ou les transitions de phase liquide-vapeur.

Stability of incompressible current-vortex sheets

“…Unlike [13], we consider the case of zero initial data for the linearized problem but introduce source terms to make the interior equations and the boundary conditions inhomogeneous because this is needed to attack the nonlinear problem. The assumption that the initial data are zero is usual and we postpone the case of non-zero initial data to the nonlinear analysis (construction of a so-called approximate solution, etc., see [4,16]). It should be noted that we have to introduce a source term also in the incompressibility condition because in the future nonlinear analysis we intend to go outside the class of divergence-free velocity fields while using the Nash-Moser technique.…”

“…That is, it is not a real mistake when we say in Section 1 that in this paper we derive estimates in usual Sobolev spaces. We note that a usual procedure towards the proof of a nonlinear existence theorem by the Nash-Moser method (see, e.g., [3,4,16]) provides for the derivation of a basic L 2 estimate for constant coefficients (like estimate (27)), the carrying this estimate over variable coefficients, and then the derivation of a tame estimate in Sobolev spaces (for variable coefficients). Of course, the a priori estimate (28) is not a tame estimate.…”

“…As for shock waves [2,8,15] and compressible vortex and current-vortex sheets [3,4,13,16], for our problem under assumption (9) the symbol associated to the front is elliptic, and this enables us to gain one derivative for the front perturbation.…”

Weakly nonlinear surface waves and subsonic phase boundaries

“…7], even though surface waves signal a failure of the so-called uniform Kreiss-Lopatinskiȋ condition, their existence is still compatible with the well-posedness of constant-coefficients linear homogeneous boundary value problems, such as (2.9). For non-linear problems, the resolution of which relies on nonhomogeneous linear problems, surface waves are responsible for a loss of regularity, see in particular the work of Coulombel and Secchi [13].…”

“…for water in extreme conditions or for superfluids). The well-posedness of the full non-linear problem in this situation has been proved by Coulombel and Secchi [13].…”