The artificial compressibility approximation for MHD equations in unbounded domain

TL;DR

This paper develops an artificial compressibility method for approximating weak solutions of incompressible MHD equations in unbounded domains, using dispersive estimates to prove convergence.

Contribution

It introduces a hyperbolic artificial compressibility approach tailored for MHD systems and proves strong convergence of the approximations.

Findings

Convergence of the approximating sequences to weak solutions.

Strong convergence of solenoidal components of velocity and magnetic fields.

Use of dispersive Strichartz estimates for analysis.

Abstract

In this paper we analyze a method of to approximation for the weak solutions of the incompressible magnetohydrodynamic equations (MHD) in unbounded domains. In particular we describe an hyperbolic version of the so called artificial compressibility method adapted to the MHD system. By exploiting the wave equation structure of the approximating system we achieve the convergence of the approximating sequences by means of dispersive estimate of Strichartz type. We prove that the soleinoidal component of the approximating velocity and magnetic fields is relatively compact and converges strongly to a weak solution of the MHD equation.