Well-posedness for Moving Interfaces with Surface Tension in Ideal Compressible MHD

TL;DR

This paper proves the local well-posedness of a free boundary problem in ideal compressible MHD with surface tension, showing that surface tension stabilizes the interface evolution and removing previous non-collinearity restrictions.

Contribution

It establishes the local existence and uniqueness of solutions for the moving interface problem in ideal compressible MHD with surface tension, using a Nash--Moser scheme and regularization techniques.

Findings

Surface tension stabilizes the vacuum interface in ideal compressible MHD.

The non-collinearity condition is unnecessary with surface tension.

The problem is well-posed under the introduced regularization and analytical framework.

Abstract

We study the local well-posedness for an interface with surface tension that separates a perfectly conducting inviscid fluid from a vacuum. The fluid flow is governed by the equations of three-dimensional ideal compressible magnetohydrodynamics (MHD), while the vacuum magnetic and electric fields are supposed to satisfy the pre-Maxwell equations. The fluid and vacuum magnetic fields are tangential to the interface. This renders a nonlinear hyperbolic-elliptic coupled problem with a characteristic free boundary. We introduce some suitable regularization to establish the solvability and tame estimates for the linearized problem. Combining the linear well-posedness result with a modified Nash--Moser iteration scheme, we prove the local existence and uniqueness of solutions of the nonlinear problem. The non-collinearity condition required by Secchi and Trakhinin [Nonlinearity 27(1): 105--169, 2014] for the case of zero surface tension becomes unnecessary in our result, which verifies the stabilizing effect of surface tension on the evolution of moving vacuum interfaces in ideal compressible MHD.