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On local solutions to a free boundary problem for incompressible viscous magnetohydrodynamics in the $L_p$-approach

Volume 566 / 2021

Yoshihiro Shibata, Wojciech M. Zajączkowski Dissertationes Mathematicae 566 (2021), 1-102 MSC: Primary 35A01; Secondary 35Q30, 35R35. DOI: 10.4064/dm777-787-3-2021 Published online: 8 September 2021


We consider the motion of incompressible magnetohydrodynamics (mhd) with resistivity in a domain bounded by a free surface which is coupled through the free surface with an electromagnetic field generated by a magnetic field prescribed on an exterior fixed boundary. On the free surface, transmission conditions for electromagnetic fields are imposed. We can distinguish two cases which have an essential influence on the proofs of existence: no jump of the magnetic field (Part 2) and a jump (Part 1). In the no jump case we prove local existence of solutions such that the velocity and the magnetic field belong to $W_r^{2,1}$, $r \gt 3$. In the case of any jump of the magnetic field we prove existence of local solutions such that the velocity belongs to $W_r^{3,3/2}$ and the magnetic field to $W_r^{2,1}$, $r \gt 5/2$.


  • Yoshihiro ShibataDepartment of Mathematics and Research Institute
    of Science and Engineering
    Waseda University
    3-4-1 Ohkubo, Shinjuku-ku
    Tokyo 169-8555, Japan
  • Wojciech M. ZajączkowskiInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    00-656 Warszawa, Poland
    Institute of Mathematics and Cryptology
    Cybernetics Faculty
    Military University of Technology
    Kaliskiego 2
    00-908 Warszawa, Poland

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