Sharp decay estimates and asymptotic stability for incompressible MHD equations without viscosity or magnetic diffusion

Abstract

Whether the global existence and uniqueness of strong solutions to n-dimensional incompressible magnetohydrodynamic (MHD for short) equations with only kinematic viscosity or magnetic diffusion holds true or not remains an outstanding open problem. In recent years, stared from the pioneer work by Lin and Zhang (Commun Pure Appl Math 67(4):531–580, 2014), much more attention has been paid to the case when the magnetic field close to an equilibrium state (the background magnetic field for short). Specifically, when the background magnetic field satisfies the Diophantine condition (see (1.2) for details), Chen et al. (Sci China Math 41:1–10, 2022) first studied the perturbation system and established the decay estimates and asymptotic stability of its solutions in 3D periodic domain \(\mathbb {T}^3\), which was then improved to \(H^{(3+2\beta )r+5+(\alpha +2\beta )}(\mathbb {T}^2)\) for 2D periodic domain \(\mathbb {T}^2\) and any \(\alpha >0\), \(\beta >0\) by Zhai (J Differ Equ 374:267–278, 2023). In this paper, we seek to find the optimal decay estimates and improve the space where the global stability is taking place. Through deeply exploring and effectively utilizing the structure of perturbation system, we discover a new dissipative mechanism, which enables us to establish the decay estimates in the Sobolev spaces with much lower regularity. Based on the above discovery, we greatly reduce the initial regularity requirement of aforesaid two works from \(H^{4r+7}(\mathbb {T}^3)\) and \(H^{(3+2\beta )r+5+(\alpha +2\beta )}(\mathbb {T}^2)\) to \(H^{(3r+3)^+}(\mathbb {T}^n)\) for \(r>n-1\) when \(n=3\) and \(n=2\) respectively. Additionally, we first present the linear stability result via the method of spectral analysis in this paper. From which, the decay estimates obtained for the nonlinear system can be seen as sharp in the sense that they are in line with those for the linearized system.