Global Solutions and Asymptotic Behavior for the Three-dimensional Viscous Non-resistive MHD System with Some Large Perturbations

Abstract

We revisit the global existence of solutions with some large perturbations to the incompressible, viscous, and non-resistive MHD system in a three-dimensional periodic domain, where the impressed magnetic field satisfies the Diophantine condition, and the intensity of the impressed magnetic field, denoted by $m$, is large compared to the perturbations. It was proved by Jiang--Jiang that the highest-order derivatives of the velocity increase with $m$, and the convergence rate of the nonlinear system towards a linearized problem is of $m^{-1/2}$ in [F. Jiang and S. Jiang, Arch. Ration. Mech. Anal., 247 (2023), 96]. In this paper, we adopt a different approach by leveraging vorticity estimates to establish the highest-order energy estimate. This strategy prevents the appearance of terms that grow with $m$, and thus the increasing behavior of the highest-order derivatives of the velocity with respect to $m$ does not appear. Additionally, we use the vorticity estimate to demonstrate the convergence rate of the nonlinear system towards a linearized problem as time or $m$ approaches infinity. Notably, our analysis reveals that the convergence rate in $m$ is faster compared to the finding of Jiang--Jiang. Finally, a key contribution of our work is the identification of an integrable time-decay of the lower dissipation, which can replace the time-decay of lower energy in closing the highest-order energy estimate. This finding significantly relaxes the regularity requirements for the initial perturbations.