On the motion of free interface in ideal incompressible MHD

TL;DR

This paper establishes a priori Sobolev estimates for smooth solutions of the free boundary problem in 3D ideal incompressible MHD, using geometric methods and a reformulation with virtual particles to handle the moving interface.

Contribution

It introduces a novel geometric approach and a fixed boundary reformulation with virtual particles for analyzing free boundary MHD problems without restrictions on magnetic field size.

Findings

A priori Sobolev estimates for free boundary MHD solutions.

Bounded covariant derivatives of magnetic fields in vacuum and on interface.

Estimates of electric field curl in vacuum essential for elliptic analysis.

Abstract

For the free boundary problem of the plasma-vacuum interface to three-dimensional ideal incompressible magnetohydrodynamics (MHD), the a priori estimates of smooth solutions are proved in Sobolev norms by adopting a geometrical point of view and some quantities such as the second fundamental form and the velocity of the free interface are estimated. In the vacuum region, the magnetic fields are described by the div-curl system of pre-Maxwell dynamics, while at the interface the total pressure is continuous and the magnetic fields are tangent to the interface, but we do not need any restrictions on the size of the magnetic fields on the free interface. We introduce the "virtual particle" endowed with a virtual velocity field in vacuum to reformulate the problem to a fixed boundary problem under the Lagrangian coordinates. The $L^2$-norms of any order covariant derivatives of the magnetic fields both in vacuum and on the boundaries are bounded in terms of initial data and the second fundamental forms of the free interface and the rigid wall. The estimates of the curl of the electric fields in vacuum are also obtained, which are also indispensable in elliptic estimates.