Nonlinear Stability of MHD Contact Discontinuities with Surface Tension

TL;DR

This paper proves the nonlinear stability of contact discontinuities in three-dimensional ideal compressible magnetohydrodynamics with surface tension, using a modified Nash--Moser scheme and Sobolev space estimates.

Contribution

It establishes the nonlinear structural stability of MHD contact discontinuities with surface tension in 3D, introducing a new regularization preserving key structural properties.

Findings

Proves nonlinear stability in Sobolev spaces.

Develops a regularization method preserving transport structure.

Derives tame estimates for the linearized problem.

Abstract

We consider the motion of two inviscid, compressible, and electrically conducting fluids separated by an interface across which there is no fluid flow in the presence of surface tension. The magnetic field is supposed to be nowhere tangential to the interface. This leads to the characteristic free boundary problem for contact discontinuities with surface tension in three-dimensional ideal compressible magnetohydrodynamics (MHD). We prove the nonlinear structural stability of MHD contact discontinuities with surface tension in Sobolev spaces by a modified Nash--Moser iteration scheme. The main ingredient of our proof is deriving the resolution and tame estimate of the linearized problem in usual Sobolev spaces of sufficiently large regularity. In particular, for solving the linearized problem, we introduce a suitable regularization that preserves the transport-type structure for the linearized entropy and divergence of the magnetic field.