A priori estimates for free boundary problem of incompressible inviscid magnetohydrodynamic flows

TL;DR

This paper establishes a priori Sobolev norm estimates for free boundary incompressible inviscid MHD flows in 2D and 3D, using geometric methods and identifying conditions for well-posedness related to pressure derivatives.

Contribution

It provides the first a priori Sobolev estimates for free boundary incompressible inviscid MHD flows in all physical dimensions, extending geometric techniques from fluid dynamics.

Findings

Identified the well-posedness condition involving the pressure derivative on the free boundary.

Established Sobolev norm bounds for the free boundary MHD problem in 2D and 3D.

Linked the pressure condition to the physical Taylor sign condition.

Abstract

In the present paper, we prove the a priori estimates of Sobolev norms for a free boundary problem of the incompressible inviscid MHD equations in all physical spatial dimensions $n=2$ and 3 by adopting a geometrical point of view used in Christodoulou-Lindblad CPAM 2000, and estimating quantities such as the second fundamental form and the velocity of the free surface. We identify the well-posedness condition that the outer normal derivative of the total pressure including the fluid and magnetic pressures is negative on the free boundary, which is similar to the physical condition (Taylor sign condition) for the incompressible Euler equations of fluids.