TY - JOUR

T1 - Local well-posedness for free boundary problem of viscous incompressible magnetohydrodynamics

AU - Oishi, Kenta

AU - Shibata, Yoshihiro

N1 - Publisher Copyright:
© 2021 by the authors. Licensee MDPI, Basel, Switzerland.

PY - 2021/3/1

Y1 - 2021/3/1

N2 - 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.

AB - 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.

KW - Free boundary problem

KW - L-L maximal regularity

KW - Local wellposedness

KW - Magnetohydorodynamics

KW - Transmission condition

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U2 - 10.3390/math9050461

DO - 10.3390/math9050461

M3 - Article

AN - SCOPUS:85102290386

VL - 9

SP - 1

EP - 33

JO - Mathematics

JF - Mathematics

SN - 2227-7390

IS - 5

M1 - 461

ER -