Local well-posedness for free boundary problem of viscous incompressible magnetohydrodynamics

Kenta Oishi, Yoshihiro Shibata*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. An electromagnetic field generated by some currents in an external domain keeps an MHD flow in a bounded domain. On the free surface, free boundary conditions for MHD flow and transmission conditions for electromagnetic fields are imposed. We proved the local well-posedness in the general setting of domains from a mathematical point of view. The solutions are obtained in an anisotropic space H1 p((0, T), H1 q) ∩ Lp((0, T), H3 q) for the velocity field and in an anisotropic space H1 p((0, T), Lq) ∩ Lp((0, T), H2 q) for the magnetic fields with 2 < p < ∞, N < q < ∞ and 2/p + N/q < 1. To prove our main result, we used the Lp-Lq maximal regularity theorem for the Stokes equations with free boundary conditions and for the magnetic field equations with transmission conditions, which have been obtained by Frolova and the second author.

Original languageEnglish
Article number461
Pages (from-to)1-33
Number of pages33
Issue number5
Publication statusPublished - 2021 Mar 1


  • Free boundary problem
  • L-L maximal regularity
  • Local wellposedness
  • Magnetohydorodynamics
  • Transmission condition

ASJC Scopus subject areas

  • Mathematics(all)


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