This paper aims to establish the global well-posedness of the free boundary problem for the incompressible viscous resistive magnetohydrodynamic 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 Ḃp,1−1+N/p(R+N) is sufficiently small, where p ∈ N, 2N − 1). Building upon prior work such as Danchin and Mucha [J. Funct. Anal. 256, 881–927 (2009) and Ogawa and Shimizu J. Differ. Equ. 274, 613–651 (2021) in the half-space setting, we establish maximal L1-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.
Zhang et al. (Wed,) studied this question.