IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i5p461-d504994.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/5/461/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/5/461/
    Download Restriction: no
    ---><---

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2021:i:5:p:461-:d:504994. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.