Well-Posedness for the Free-Boundary Ideal Compressible Magnetohydrodynamic Equations with Surface Tension

Abstract

We establish the local existence and uniqueness of solutions to the free-boundary ideal compressible magnetohydrodynamic equations with surface tension in three spatial dimensions by a suitable modification of the Nash--Moser iteration scheme. The main ingredients in proving the convergence of the scheme are the tame estimates and unique solvability of the linearized problem in the anisotropic Sobolev spaces $H_{\ast }^m$ for $m$ large enough. In order to derive the tame estimates, we make full use of the boundary regularity enhanced from the surface tension. The unique solution of the linearized problem is constructed by designing some suitable $\epsilon$--regularization and passing to the limit $\epsilon \to 0$.

To appear in: Mathematische Annalen

References in corpus (2)

• The Magnetic Rayleigh-Taylor Instability in Three Dimensions
• Nonlinear Evolution of the Magnetohydrodynamic Rayleigh-Taylor Instability

Cited by in corpus (4)

• Zero Surface Tension Limit of the Free-Boundary Problem in Incompressible Magnetohydrodynamics
• Well-posedness for Moving Interfaces with Surface Tension in Ideal Compressible MHD
• Nonlinear Stability of MHD Contact Discontinuities with Surface Tension
• Anisotropic Regularity of the Free-Boundary Problem in Compressible Ideal Magnetohydrodynamics