On the motion of free interface in two-dimensional ideal incompressible MHD

TL;DR

This paper establishes a priori estimates for the free boundary problem in 2D ideal incompressible MHD, analyzing the magnetic field and interface dynamics using a geometric approach.

Contribution

It provides the first rigorous Sobolev norm estimates for the plasma-vacuum interface in 2D ideal incompressible MHD, incorporating geometric analysis of the interface and magnetic fields.

Findings

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

Continuity of total pressure and tangency of magnetic field at the interface.

Estimates depend on initial data and geometric properties of the interface.

Abstract

For the free boundary problem of the plasma-vacuum interface to ideal incompressible magnetohydrodynamics (MHD) in two-dimensional space, the a priori estimates of solutions are proved in Sobolev norms by adopting a geometrical point of view. In the vacuum region, the magnetic field is described by the div-curl system of pre-Maxwell dynamics, while at the interface the total pressure is continuous and the magnetic field is tangent to the boundary. We prove that the $L^2$ norms of any order covariant derivatives of the magnetic field 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.