Anisotropic Regularity of the Free-Boundary Problem in Compressible Ideal Magnetohydrodynamics
We consider 3D free-boundary compressible ideal magnetohydrodynamic (MHD) system under the Rayleigh-Taylor sign condition. It describes the motion of a free-surface perfect conducting fluid in an electro-magnetic field. A local existence and uniqueness result was recently proved by Trakhinin and Wang [64] by using Nash-Moser iteration. However, that result loses regularity going from data to solution. In this paper, we show that the Nash-Moser iteration scheme in [64] can be improved such that the local-in-time smooth solution exists and is unique when the initial data is smooth and satisfies the compatibility condition up to infinite order. Second, we prove the a priori estimates without loss of regularity for the free-boundary compressible MHD system in Lagrangian coordinates in anisotropic Sobolev space, with more regularity tangential to the boundary than in the normal direction. It is based on modified Alinhac good unknowns, which take into account the covariance under the change of coordinates to avoid the derivative loss; full utilization of the cancellation structures of MHD system, to turn normal derivatives into tangential ones; and delicate analysis in anisotropic Sobolev spaces. As a result, we can prove the uniqueness and the continuous dependence on initial data provided the local existence, and a continuation criterion for smooth solution. Finally, we extend the local well-posedness theorem to the case of initial data only satisfying compatibility conditions up to finite order, assuming these can be approximated by data satisfying infinitely many compatibility conditions.