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Local Well-Posedness for the Magnetohydrodynamics in the Different Two Liquids Case

Author

Listed:
  • Elena Frolova

    (St. Petersburg State Electrotechnical University «LETI», Prof. Popova 5, 191126 St. Petersburg, Russia
    Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetskii pr. 28, Peterhof, 198504 St. Petersburg, Russia)

  • Yoshihiro Shibata

    (Department of Mathematics, Waseda University, Tokyo 169-8555, Japan
    Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA 15261, USA)

Abstract

We consider the free boundary problem of MHD in the multi-dimensional case. This problem describes the motion of two incompressible fluids separated by a closed interface under the action of a magnetic field. This problem is overdetermined, and we find an equivalent system of equations which is uniquely solvable locally in time in the L p - L q maximal regularity class, where 1 < p , q < ∞ and 2 / p + N / q < 1 . As a result, the original two-phase problem for the MHD is solvable locally in time.

Suggested Citation

  • Elena Frolova & Yoshihiro Shibata, 2022. "Local Well-Posedness for the Magnetohydrodynamics in the Different Two Liquids Case," Mathematics, MDPI, vol. 10(24), pages 1-45, December.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:24:p:4751-:d:1003169
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