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). Dans ce document, 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 $t>0$. En outre, 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).

Cet article explore les excursions dans le temps et leurs implications.

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