Well-posedness for the Linearized Free Boundary Problem of Incompressible Ideal Magnetohydrodynamics Equations

TL;DR

This paper establishes the local-in-time well-posedness of the linearized free boundary problem for incompressible ideal magnetohydrodynamics equations in a bounded domain, using energy estimates and Lie derivatives.

Contribution

It introduces a novel approach to linearize and analyze the free boundary MHD problem by expressing the magnetic field via the velocity and deformation tensors, and proves well-posedness.

Findings

Proves local-in-time well-posedness of the linearized problem.

Develops energy estimates using tangential derivatives and curl.

Utilizes Lie derivatives and smooth-out approximation techniques.

Abstract

We study the well-posedness theory for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations in a bounded domain. We express the magnetic field in terms of the velocity field and the deformation tensors in the Lagrangian coordinates, and substitute the magnetic field into the momentum equation to get an equation of the velocity in which the initial magnetic field serves only as a parameter. Then, we linearize this equation with respect to the position vector field whose time derivative is the velocity, and obtain the local-in-time well-posedness of the solution by using energy estimates of the tangential derivatives and the curl with the help of Lie derivatives and the smooth-out approximation.