Global well-posedness of the free boundary problem for incompressible viscous resistive MHD in critical Besov spaces

TL;DR

This paper proves the global existence and uniqueness of solutions for the free boundary incompressible viscous resistive MHD equations in critical Besov spaces, extending previous results with new maximal regularity estimates.

Contribution

It establishes global well-posedness in critical Besov spaces for free boundary MHD, utilizing maximal regularity and contraction mapping techniques.

Findings

Global well-posedness in critical Besov spaces

Maximal L^1-regularity for linearized equations

Existence and uniqueness via contraction mapping

Abstract

This paper aims to establish the global well-posedness of the free boundary problem for the incompressible viscous resistive magnetohydrodynamic (MHD) equations. Under the framework of Lagrangian coordinates, a unique global solution exists in the half-space provided that the norm of the initial data in the critical homogeneous Besov space $\dot{B}_{p, 1}^{-1+N/p}(\mathbb{R}_{+}^N)$ is sufficiently small, where $p \in [N, 2N-1)$. Building upon prior work such as (Danchin and Mucha, J. Funct. Anal. 256 (2009) 881--927) and (Ogawa and Shimizu, J. Differ. Equations 274 (2021) 613--651) in the half-space setting, we establish maximal $L^{1}$-regularity for both the Stokes equations without surface stress and the linearized equations of the magnetic field with zero boundary condition. The existence and uniqueness of solutions to the nonlinear problems are proven using the Banach contraction mapping principle.