A global stability result for incompressible magnetohydrodynamics

TL;DR

This work proves a global stability result for the homogeneous, incompressible MHD system on the torus $\mathbf{T}^d$ with positive viscosity $\nu$ and resistivity $\eta$, in a $C^\infty$ setting and with fully explicit $H^p$-norm estimates. The authors develop an a posteriori framework based on approximate solutions, recasting the MHD equations via a linear operator $\mathscr{A}$ and a bilinear map $\mathscr{P}$, and establish quantitative control inequalities that guarantee global existence and decay of perturbations around a given global decaying solution $\mathbf{v}$. A key contribution is a global stability theorem that provides explicit bounds on the distance between nearby solutions and the decaying reference solution, with constants determined by Sobolev norms of $\mathbf{v}$ and integrals of its norms in time. The paper also introduces generalized Beltrami pairs as large-data initial configurations that yield global, decaying MHD solutions, and shows these data can have arbitrarily large $H^p$-norms, enriching the landscape of explicitly controllable MHD dynamics.

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.