Global stability and asymptotic behavior for the incompressible MHD equations without viscosity or magnetic diffusion

TL;DR

This paper rigorously proves that a background magnetic field stabilizes incompressible MHD flows without viscosity or magnetic diffusion, providing explicit decay rates and relaxing initial data regularity requirements across all Sobolev norms.

Contribution

It offers a dimension-independent analytical framework that establishes global stability and decay rates for the incompressible MHD equations with partial diffusion, under Diophantine conditions.

Findings

Global stability of solutions established

Explicit decay rates derived for perturbations

Relaxed regularity conditions on initial data

Abstract

Physical experiments and numerical simulations have revealed a remarkable stabilizing phenomenon: a background magnetic field stabilizes and dampens electrically conducting fluids. This paper provides a rigorous mathematical justification of this effect for the $n$-dimensional incompressible magnetohydrodynamic equations with partial diffusion on periodic domains. We establish the global stability and derive explicit decay rates for perturbations around an equilibrium magnetic field satisfying the Diophantine condition. Our results yield the \textit{effective decay rates in all intermediate Sobolev norms} and \textit{significantly relax the regularity requirements} on the initial data compared with previous works (\textit{Sci. China Math.} 41:1--10, 2022; \textit{J. Differ. Equ.} 374:267--278, 2023; \textit{Calc. Var. Partial Differ. Equ.} 63:191, 2024). Furthermore, the analytical framework developed here is dimension-independent and can be flexibly adapted to other fluid models with partial dissipation.