A global stability result for incompressible magnetohydrodynamics

TL;DR

This paper establishes a comprehensive global stability result for homogeneous, incompressible magnetohydrodynamics equations on a torus across multiple dimensions, providing explicit Sobolev norm estimates for smooth solutions with decay over time.

Contribution

It extends the stability analysis of Navier-Stokes equations to MHD, offering explicit estimates and a new class of large initial data leading to global solutions.

Findings

Proves global stability for MHD equations with positive viscosity and resistivity.

Provides explicit Sobolev norm estimates for solutions.

Introduces a class of large initial data yielding global, decaying solutions.

Abstract

We propose a result of global stability for the equations of homogeneous, incompressible magnetohydrodynamics (MHD) on a torus of any dimension $d \in \{2,3,...\}$, with positive viscosity and resistivity. This result applies to the $C^\infty$ global solutions, with a conveniently defined decay property for large times; it is expressed by fully explicit estimates, formulated via $H^p$-type Sobolev norms of arbitrarily high order $p$. The present stability result is similar to that proposed by one of us for the Navier-Stokes (NS) equation \cite{glosta}; it is derived from a suitable formulation of the MHD equations proposed in our previous work \cite{MHD}, emphasizing strong structural analogies with the NS case. A basic tool in the proof of the present stability result is a general theory of approximate solutions of the MHD Cauchy problem, that we developed in \cite{MHD} on the grounds of previous results on the NS equation \cite{smooth} and of the above structural similarities. We also introduce a class of Beltrami-type initial data for the MHD equations; although being arbitrarily large, these data produce global and decaying MHD solutions, fitting the framework of the present stability result. Comparisons with the previous literature on these subjects are performed.