Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Small Alfvén Number Limit for the Global-in-time Solutions of Incompressible MHD Equations with General Initial Data

Yuan Cai, Xiufang Cui, Fei Jiang, Hao Liu

TL;DR

The paper addresses the global-in-time behavior of incompressible MHD with a small Alfvén number $\varepsilon$ in $\mathbb{R}^n$ ($n=2,3$) for general initial data. By recasting the system in Elsässer variables and employing a ghost-weight energy method, the authors prove uniform-in-$\varepsilon$ global existence for a transformed system and derive an explicit decay framework. They also establish an error estimate showing nonlinear interactions vanish in the small-$\varepsilon$ limit when compared to the linear problem, and they analyze the dissipation-vanishing and non-dissipative limits, with pointwise decay results for the MHD fields. The results extend prior local-in-time and ideal-MHD limits to a global-in-time, viscous-resistive setting and illuminate how strong magnetic fields suppress nonlinear interactions as $\varepsilon\to0$, providing quantitative convergence and decay rates suitable for SEO and downstream analyses.

Abstract

The small Alfvén 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 Alfvén 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 Alfvén numbers, and thus extend Bardos et al.'s results of the \emph{ideal} MHD equations in (Trans Am Math Soc 305: 175--191, 1988).

Small Alfvén Number Limit for the Global-in-time Solutions of Incompressible MHD Equations with General Initial Data

TL;DR

The paper addresses the global-in-time behavior of incompressible MHD with a small Alfvén number in () for general initial data. By recasting the system in Elsässer variables and employing a ghost-weight energy method, the authors prove uniform-in- global existence for a transformed system and derive an explicit decay framework. They also establish an error estimate showing nonlinear interactions vanish in the small- limit when compared to the linear problem, and they analyze the dissipation-vanishing and non-dissipative limits, with pointwise decay results for the MHD fields. The results extend prior local-in-time and ideal-MHD limits to a global-in-time, viscous-resistive setting and illuminate how strong magnetic fields suppress nonlinear interactions as , providing quantitative convergence and decay rates suitable for SEO and downstream analyses.

Abstract

The small Alfvén number (denoted by ) 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 (i.e. a spatially periodic domain) tend to a solution of 2D ideal MHD equations in the distribution sense as by Schochet's fast averaging method in (J. Differential Equations, 114: 476--512, 1994). In this paper, we revisit the small Alfvén number limit in with , , 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 for any given time-space variable with . 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 Alfvén numbers, and thus extend Bardos et al.'s results of the \emph{ideal} MHD equations in (Trans Am Math Soc 305: 175--191, 1988).

Paper Structure

This paper contains 11 sections, 5 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Reformulation
  4. Main results
  5. Results for the transformed system \ref{['1.4']}
  6. Results for the original system \ref{['1sa.1']} with $\tilde{\mu}=\nu$
  7. An additional result for dissipation vanishing
  8. Existence of solutions with uniform-in-$\varepsilon$ estiamtes
  9. An error estimate for nonlinear interactions vanishing
  10. Decay of solutions
  11. Non-dissipative limit

Key Result

Theorem 2.1

Let $n=2$, $3$, $\nu\geqslant 0$, $\varepsilon\nu\leqslant 1/2$, $1<2s <4/{3}$, $\Lambda_\pm^{0} \in H^k_\sigma$ with the integer $k\geqslant4$ and There exist a small constant $c_1\in(0,1)$ and large constant $c_2$, which depend only on $n$, $k$ and ${s}$ such that, for any $\varepsilon$ satisfying the Cauchy problem 1.4 defined on $\mathbb{R}^n\times \mathbb{R}_0^+$ admits a unique global-in-t

Theorems & Definitions (5)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5