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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.16650 |
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| _version_ | 1866908334398898176 |
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| author | Cai, Yuan Cui, Xiufang Jiang, Fei Liu, Hao |
| author_facet | Cai, Yuan Cui, Xiufang Jiang, Fei Liu, Hao |
| contents | 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). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_16650 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Small Alfvén Number Limit for the Global-in-time Solutions of Incompressible MHD Equations with General Initial Data Cai, Yuan Cui, Xiufang Jiang, Fei Liu, Hao Analysis of PDEs 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). |
| title | Small Alfvén Number Limit for the Global-in-time Solutions of Incompressible MHD Equations with General Initial Data |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.16650 |