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We consider 3D free-boundary compressible ideal magnetohydrodynamic (MHD)system under the Rayleigh-Taylor sign condition. It describes the motion of afree-surface perfect conducting fluid in an electro-magnetic field. A localexistence and uniqueness result was recently proved by Trakhinin and Wang [64]by using Nash-Moser iteration. However, that result loses regularity going fromdata to solution. In this paper, we show that the Nash-Moser iteration schemein [64] can be improved such that the local-in-time smooth solution exists andis unique when the initial data is smooth and satisfies the compatibilitycondition up to infinite order. Second, we prove the a priori estimates withoutloss of regularity for the free-boundary compressible MHD system in Lagrangiancoordinates in anisotropic Sobolev space, with more regularity tangential tothe boundary than in the normal direction. It is based on modified Alinhac goodunknowns, which take into account the covariance under the change ofcoordinates to avoid the derivative loss; full utilization of the cancellationstructures of MHD system, to turn normal derivatives into tangential ones; anddelicate analysis in anisotropic Sobolev spaces. As a result, we can prove theuniqueness and the continuous dependence on initial data provided the localexistence, and a continuation criterion for smooth solution. Finally, we extendthe local well-posedness theorem to the case of initial data only satisfyingcompatibility conditions up to finite order, assuming these can be approximatedby data satisfying infinitely many compatibility conditions.

Anisotropic Regularity of the Free-Boundary Problem in Compressible Ideal Magnetohydrodynamics