Strong solutions to the three-dimensional two-phase magnetohydrodynamic equations

TL;DR

This paper proves the existence of strong solutions for the three-dimensional two-phase magnetohydrodynamic equations in bounded domains, accounting for surface tension and magnetic effects, using a fixed-point approach after reformulating the problem.

Contribution

It introduces a reformulation via the Hanzawa transformation and establishes strong solution existence for complex MHD equations with free interfaces in 3D.

Findings

Existence of strong solutions in bounded domains.

Effective reformulation of free interface problem.

Handling nonlinearities with fixed-point method.

Abstract

In this paper, we study the existence of strong solutions to the two-phase magnetohydrodynamic equations in a bounded domain $\Omega\subseteq \mathbb{R}^3$. The fluids are incompressible, viscous, and resistive. The surface tension is considered. The equations are reformulated using the Hanzawa transformation, which turns the free interface into a fixed one for a short time. The study of the new equations is then divided into the principal part and the nonlinear part. Due to the effect of the magnetic field and the complexity of the transformation in generic bounded domains, the Fr\'echet derivatives of nonlinearities have to be carefully estimated. The equations can then be solved using the fixed-point argument by finding a contraction mapping, which follows the estimates of the nonlinear part.