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

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 $\mathit{\text{effective}}\mathit{\text{decay}}\mathit{\text{rates}}\mathit{\text{in}}\mathit{\text{all}}\mathit{\text{intermediate}}\mathit{\text{Sobolev}}\mathit{\text{norms}}$ and $\mathit{\text{significantly}}\mathit{\text{relax}}\mathit{\text{the}}\mathit{\text{regularity}}\mathit{\text{requirements}}$ on the initial data compared with previous works ($\mathit{\text{Sci.}}\mathit{\text{China}}\mathit{\text{Math.}}$ 41:1–10, 2022; $\mathit{\text{J.}}\mathit{\text{Differ.}}\mathit{\text{Equ.}}$ 374:267–278, 2023; $\mathit{\text{Calc.}}\mathit{\text{Var.}}\mathit{\text{Partial}}\mathit{\text{Differ.}}\mathit{\text{Equ.}}$ 63:191, 2024). Furthermore, the analytical framework developed here is dimension-independent and can be flexibly adapted to other fluid models with partial dissipation.