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Local Well-Posedness for Free Boundary Problem of Viscous Incompressible Magnetohydrodynamics

Author

Listed:
  • Kenta Oishi

    (Department of Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, Aichi 464-8602, Japan)

  • Yoshihiro Shibata

    (Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA 15260, USA
    Department of Mathematics, Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan
    Partially supported by Top Global University Project, JSPS Grant-in-aid for Scientific Research (A) 17H0109, and Toyota Central Research Institute Joint Research Fund.)

Abstract

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 H p 1 ( ( 0 , T ) , H q 1 ) ∩ L p ( ( 0 , T ) , H q 3 ) for the velocity field and in an anisotropic space H p 1 ( ( 0 , T ) , L q ) ∩ L p ( ( 0 , T ) , H q 2 ) for the magnetic fields with 2 < p < ∞ , N < q < ∞ and 2 / p + N / q < 1 . To prove our main result, we used the L p - L q 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.

Suggested Citation

  • Kenta Oishi & Yoshihiro Shibata, 2021. "Local Well-Posedness for Free Boundary Problem of Viscous Incompressible Magnetohydrodynamics," Mathematics, MDPI, vol. 9(5), pages 1-33, February.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:5:p:461-:d:504994
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