Yanping Zhou | Searcharxiv

arXiv2025

Global stability and asymptotic behavior for incompressible ideal MHD equations with velocity damping term

In this article, we study the stability and large time behavior for an multi-dimensional incompressible magnetohydrodynamical system with a velocity damping term, for small perturbations near a steady-state of magnetic field fulfilling the Diophantine condition. Our results mathematically characterize the background magnetic field exerts the stabilizing effect, and bridge the gap left by previous work with respect to the asymptotic behavior in time. Our proof approach mainly relies on the Fourier analysis and energy estimates. In addition, we provide a versatile analytical framework applicable to many other partially dissipative fluid models.

math.AP↗

arXiv2025

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 \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.

math.AP↗