Asymptotic Behaviors of Global Solutions to the Two-Dimensional Non-resistive MHD Equations with Large Initial Perturbations

TL;DR

This paper investigates the long-term behavior of strong solutions to 2D non-resistive MHD equations with large initial perturbations, revealing decay rates and convergence properties influenced by magnetic field strength.

Contribution

It establishes existence, uniqueness, decay rates, and convergence results for strong solutions under large initial perturbations, extending analysis techniques to the inviscid case.

Findings

Velocity decays faster than magnetic field in H^2-norm.

Strong solutions converge to linearized solutions as magnetic field strength increases.

Existence and uniqueness of solutions are proven for large initial perturbations.

Abstract

This paper is concerned with the asymptotic behaviors of global strong solutions to the incompressible non-resistive viscous magnetohydrodynamic (MHD) equations with large initial perturbations in two-dimensional periodic domains in Lagrangian coordinates. First, motivated by the odevity conditions imposed in [Arch. Ration. Mech. Anal. 227 (2018), 637--662], we prove the existence and uniqueness of strong solutions under some class of large initial perturbations, where the strength of impressive magnetic fields depends increasingly on the $H^2$-norm of the initial perturbation values of both velocity and magnetic field. Then, we establish time-decay rates of strong solutions. Moreover, we find that $H^2$-norm of the velocity decays faster than the perturbed magnetic field. Finally, by developing some new analysis techniques, we show that the strong solution convergence in a rate of the field strength to the solution of the corresponding linearized problem as the strength of the impressive magnetic field goes to infinity. In addition, an extension of similar results to the corresponding inviscid case with damping is presented.