Varifold solutions of the two-phase three-dimensional magnetohydrodynamic equations

TL;DR

This paper establishes the existence of varifold solutions for the complex three-dimensional two-phase magnetohydrodynamic interface problem, incorporating surface tension and magnetic effects, using a fixed-point and weak convergence approach.

Contribution

It introduces a novel approach to prove existence of solutions for 3D two-phase MHD equations with free interfaces, employing varifold solutions and fixed-point methods.

Findings

Existence of varifold solutions for the 3D two-phase MHD problem.

Construction of approximate solutions via Galerkin method.

Application of Schauder fixed-point theorem and weak convergence techniques.

Abstract

In this paper we study the three-dimensional two-phase magnetohydrodynamic interface problem in a bounded domain. The two incompressible fluids are both Newtonian and the surface tension is considered. We shall use the Galerkin method to construct the approximate solutions in a bounded domain. Due to the magnetic field in the magnetohydrodynamic equations, we cannot use the method of monotone operators to solve the approximate equations. Instead, we will construct an iterating operator and solve the equations by finding the fixed-point of the operator. To deal with the free interface, we shall prove the compactness of the iterating operator and then use the Schauder fixed-point theorem. The existence of the varifold solution is established by the weak convergence method.