On application of the Nash–Moser method to weakly well-posed free boundary problems

We discuss the usage of the Nash-Moser method for the proof of the local-in-time existence of smooth solutions to free boundary problems whose linearizations are weakly well-posed problems. We mainly consider problems for hyperbolic systems of conservation laws, but our approach is also applicable for such systems as, for example, the incompressible Euler equations. Weak well-posedness means that the Kreiss-Lopatiski condition for the constant coefficients linearized problem holds only in a weak sense. In fact, weak well-posedness means neutral stability and usually implies the loss of derivatives phenomenon in a priori estimates for the linearized problem. The main idea of the Nash-Moser method is just the compensation of lost derivatives at each step of the iteration process for the nonlinear problem by using a sequence of smoothing operators. We briefly discuss peculiarities of the application of the Nash-Moser method to free boundary problems for the compressible Euler equations and the equations of ideal compressible magnetohydrodynamics (MHD). Our examples are the compressible liquid-vacuum problem, the plasma-vacuum interface problem and the free boundary problem for MHD contact discontinuities.