The Stability of Compressible Vortex Sheets in Two Space Dimensions

The Stability of Compressible Vortex Sheets in Two Space Dimensions JEAN-FRANC¸OIS COULOMBEL & PAOLO SECCHI ABSTRACT. We study the linear stability of compressible vortex sheets in two space dimensions. Under a supersonic condition that precludes violent instabilities, we prove an energy estimate for the linearized boundary value problem. Since the problem is characteristic, the estimate we prove exhibits a loss of control on the trace of the solution. Furthermore, the failure of the uni- form Kreiss-Lopatinskii condition yields a loss of derivatives in the energy estimate. 1. INTRODUCTION A velocity discontinuity in an inviscid flow is called a vortex sheet. In three-space dimensions, a vortex sheet has vorticity concentrated along a surface in the space. In two-space dimensions, the vorticity is concentrated along a curve in the plane. The present paper deals with compressible vortex sheets, i.e., vortex sheets in a compressible flow. If the solution is piecewise constant on either side of the interface of discon- tinuity, one has planar vortex sheets in the three dimensional case and rectilinear vortex sheets in the two dimensional case, respectively. The linear stability of pla- nar and rectilinear compressible vortex sheets has been analyzed a long time ago, see [12, 27]. In three space dimensions, planar vortex sheets are known to be vio- lently unstable (see e.

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