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Global well-posedness of the free boundary problem for incompressible viscous resistive MHD in critical Besov spaces
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  • Wei Zhang,
  • Jie Fu,
  • Chengchun Hao,
  • Siqi Yang
Wei Zhang
Hefei University
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Jie Fu
Institute of Mathematics Academy of Mathematics and Systems Science Chinese Academy of Sciences
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Chengchun Hao
Institute of Mathematics Academy of Mathematics and Systems Science Chinese Academy of Sciences

Corresponding Author:hcc@amss.ac.cn

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Siqi Yang
Institute of Mathematics Academy of Mathematics and Systems Science Chinese Academy of Sciences
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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 B _ p , 1 – 1 + N / p ( R + N ) is sufficiently small, where p∈[ N,2 N–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.
18 Jul 2024Submitted to Mathematical Methods in the Applied Sciences
18 Jul 2024Submission Checks Completed
18 Jul 2024Assigned to Editor
26 Jul 2024Review(s) Completed, Editorial Evaluation Pending
26 Jul 2024Reviewer(s) Assigned