Nonlinear stability of relativistic vortex sheets in two spatial dimensions

We study vortex sheets for the relativistic Euler equations in three-dimensional Minkowski spacetime. The problem is a nonlinear hyperbolic problem with a characteristic free boundary. The so-called Lopatinskii condition holds only in a weak sense, which yields losses of derivatives. A necessary condition for the weak stability is obtained by analyzing roots of the Lopatinskii determinant associated to the linearized problem. Under such stability condition,  we prove short-time existence and nonlinear stability of relativistic vortex sheets by the Nash-Moser iterative scheme.