Small Alfv\'en Number Limit for the Global-in-time Solutions of Incompressible MHD Equations with General Initial Data

Abstract

The small Alfv\'en number (denoted by $\varepsilon$) limit (one type of large parameter limits, i.e. singular limits) in magnetohydrodynamic (abbr. MHD) equations was first proposed by Klainerman--Majda in (Comm. Pure Appl. Math. 34: 481--524, 1981). Recently Ju--Wang--Xu mathematically verified that the \emph{local-in-time} solutions of three-dimensional (abbr. 3D) ideal (i.e. the absence of the dissipative terms) incompressible MHD equations with general initial data in $\mathbb{T}^3$ (i.e. a spatially periodic domain) tend to a solution of 2D ideal MHD equations in the distribution sense as $\varepsilon\to 0$ by Schochet's fast averaging method in (J. Differential Equations, 114: 476--512, 1994). In this paper, we revisit the small Alfv\'en number limit in $\mathbb{R}^n$ with $n=2$, $3$, and develop another approach, motivated by Cai--Lei's energy method in (Arch. Ration. Mech. Anal. 228: 969--993, 2018), to establish a new conclusion that the \emph{global-in-time} solutions of incompressible MHD equations (including the viscous resistive case) with general initial data converge to zero as $\varepsilon\to 0$ for any given time-space variable $(x,t)$ with $t>0$. In addition, we find that the large perturbation solutions and vanishing phenomenon of the nonlinear interactions also exist in the \emph{viscous resistive} MHD equations for small Alfv\'en numbers, and thus extend Bardos et al.'s results of the \emph{ideal} MHD equations in (Trans Am Math Soc 305: 175--191, 1988).