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45 pages, 523 KiB

Open AccessArticle

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

by Elena Frolova and Yoshihiro Shibata

Mathematics 2022, 10(24), 4751; https://doi.org/10.3390/math10244751 - 14 Dec 2022

Cited by 2 | Viewed by 2519

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 [...] Read more.

L_{p}

L_{q}

maximal regularity class, where

1 < p, q < \infty

and

2 / p + N / q < 1

. As a result, the original two-phase problem for the MHD is solvable locally in time. Full article

(This article belongs to the Special Issue Maximal Regularity, Stability Estimates and Mathematical Fluid Dynamics II)

33 pages, 469 KiB

Open AccessArticle

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

by Kenta Oishi and Yoshihiro Shibata

Mathematics 2021, 9(5), 461; https://doi.org/10.3390/math9050461 - 24 Feb 2021

Cited by 1 | Viewed by 1673

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}) \cap L_{p} ((0, T), H_{q}^{3})

for the velocity field and in an anisotropic space

H_{p}^{1} ((0, T), L_{q}) \cap L_{p} ((0, T), H_{q}^{2})

for the magnetic fields with

2 < p < \infty

N < q < \infty

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. Full article

(This article belongs to the Special Issue Maximal Regularity, Stability Estimates and Mathematical Fluid Dynamics)

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