Well-posedness for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations

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Detail(s)

Original languageEnglish
Pages (from-to)542-601
Journal / PublicationJournal of Differential Equations
Volume299
Online published3 Aug 2021
Publication statusPublished - 25 Oct 2021

Abstract

The well-posedness theory is studied 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 Lagrangian coordinates, and substitute it into the momentum equation to get an equation of the velocity in which the initial magnetic field serves only as a parameter. Then, the velocity equation is linearized with respect to the position vector field whose time derivative is the velocity. In this formulation, a key idea is to use the Lie derivative of the magnetic field taking the advantage that the magnetic field is tangential to the free boundary and divergence free. This paper contributes to the program of developing geometric approaches to study the well-posedness of free boundary problems of ideal magnetohydrodynamics equations under the condition of Taylor sign type for general free boundaries not restricted to graphs.

Research Area(s)

  • Free boundary problem, Incompressible ideal magnetohydrodynamics equations, Linearized equations, Local well-posedness