Nonlinear stability and existence of compressible vortex sheets in 2D elastodynamics

Abstract

The nonlinear stability and local existence of compressible vortex sheets for the two-dimensional isentropic elastic fluid are established in the usual Sobolev spaces. The problem has a characteristic free boundary, and the Kreiss–Lopatinskiĭ condition is satisfied only in a weak form. This paper completes the previous works [6,7] of the first three authors where the weakly linear stability of the rectilinear vortex sheets is proved by means of an upper triangularization technique. Our proof is based on certain higher-order energy estimates and an appropriate modification of the Nash–Moser iteration. In particular, the estimate for the normal derivatives of the characteristic variables can be recovered from that for the linearized divergences and vorticities.