Regularity results for a model in magnetohydrodynamics with imposed pressure

TL;DR

This paper investigates the regularity and existence of solutions for a magnetohydrodynamics model with pressure boundary conditions, extending known results to more general boundary settings and establishing conditions for solution uniqueness.

Contribution

It introduces regularity results and existence theorems for MHD systems with pressure boundary conditions, a setting less studied than Dirichlet boundary conditions.

Findings

Existence of weak solutions in the Hilbert space framework.

Regularity results in $W^{1,p}$ and $W^{2,p}$ spaces for certain p.

Existence and uniqueness of solutions under small data assumptions.

Abstract

The magnetohydrodynamics (MHD) problem is most often studied in a framework where Dirichlet type boundary conditions on the velocity field is imposed. In this Note, we study the (MHD) system with pressure boundary condition, together with zero tangential trace for the velocity and the magnetic field. In a three-dimensional bounded possibly multiply connected domain, we first prove the existence of weak solutions in the Hilbert case, and later, the regularity in $W^{1,p}(\Omega)$ for $p\geq 2$ and in $W^{2,p}(\Omega)$ for $p\geq 6/5$ using the regularity results for some Stokes and elliptic problems with this type of boundary conditions. Furthermore, under the condition of small data, we obtain the existence and uniqueness of solutions in $W^{1,p}(\Omega)$ for $3/2<p<2$ by using a fixed-point technique over a linearized (MHD) problem.