Well-Posedness for a Magnetohydrodynamical Model with Intrinsic Magnetisation

TL;DR

This paper establishes the local existence, uniqueness, and regularity of weak solutions for a magnetohydrodynamical model that incorporates intrinsic magnetisation and the Landau-Lifshitz-Gilbert equation.

Contribution

It provides the first rigorous mathematical analysis of well-posedness for a magnetohydrodynamical system with intrinsic magnetisation.

Findings

Proved local existence of weak solutions.

Established uniqueness under certain conditions.

Demonstrated regularity properties of solutions.

Abstract

Ferromagnetic magnetohydrodynamics concerns the study of conducting fluids with intrinsic magnetisation under the influence of a magnetic field. It is a generalisation of the magnetohydrodynamical equations and takes into account the dynamics of the magnetisation of a fluid. First proposed by Lingam (Lingam, `Dissipative effects in magnetohydrodynamical models with intrinsic magnetisation', Communications in Nonlinear Science and Numerical Simulation Vol 28, pp 223-231, 2015), the usual equations of magnetohydrodynamics, namely the Navier-Stokes equation and the induction equation, are coupled with the Landau-Lifshitz-Gilbert equation. In this paper, the local existence, uniqueness and regularity of weak solutions to this system are discussed.