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

Kenta Oishi,Yoshihiro Shibata

doi:10.3390/math9050461

Kenta Oishi, Yoshihiro Shibata

Open Access

https://doi.org/10.3390/math9050461

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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 Hp1((0,T),Hq1)∩Lp((0,T),Hq3) for the velocity field and in an anisotropic space Hp1((0,T),Lq)∩Lp((0,T),Hq2) 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.

Highlights

Accepted: 17 February 2021In this paper, we prove the local well-posedness of a free boundary problem for the viscous non-homogeneous incompressible magnetohydrodynamics
We prove the local well-posedness of a free boundary problem for the viscous non-homogeneous incompressible magnetohydrodynamics
Throughout the paper, we assume that Ω± are uniform C2 domains, that the weak Dirichlet problem is uniquely solvable in Ω+ (The definition of uniform C2 domains and the weak Dirichlet problem will be given in Section 3 below.) and that dist (Γ, S− ) ≥ 2d− with some positive constants d−, where dist( A, B) denotes the distance of any two subsets A and B of R N defined by setting dist( A, B) = inf{| x − y| | x ∈ A, y ∈ B}

Summary

Introduction

We prove the local well-posedness of a free boundary problem for the viscous non-homogeneous incompressible magnetohydrodynamics. The purpose of this paper is to prove the local well-posedness of the free boundary problem formulated by the set of the following equations: div v = 0. The initial-boundary value problem for MHD equations with non-slip conditions for the velocity vector field and perfect wall conditions for the magnetic vector field was studied by Sermange and Temam [5] in a bounded domain and by Yamaguchi [6] in an exterior domain. In their studies [5,6], the boundary is fixed.

Derivation of Nonlinear Terms

Linear Theory

The Stokes Equations with Free Boundary Conditions

Two Phase Problem for the Linear Electro-Magnetic Vector Field Equations

Estimate of Non-Linear Terms

A Proof of Theorem 1