Global Solutions to 3D Incompressible MHD System with Dissipation in Only One Direction
Global Solutions to 3D Incompressible MHD System with Dissipation in Only One Direction
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DOI：
10.1137/22m1471274
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发表时间：
2022-10
期刊：
影响因子：
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通讯作者：
Hongxia Lin;Jiahong Wu;Yi Zhu
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文献类型：
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作者：
Hongxia Lin;Jiahong Wu;Yi Zhu
含单向耗散的三维不可压Navier-Stokes方程的小数据整体适定性问题一直是一个未解决的问题。只在一个方向上的耗散，比如说$\Partial_1^2u$，根本不足以控制整个空间$\mathbb R^3$的非线性。Paicu和Zhang‘cite{ZHANG1}的漂亮工作通过观察一个关键的Poincar型不等式解决了空间域在$x1$方向有界的情况。受这一公开问题的启发，通过对背景磁场稳定效应的实验观测，本文旨在了解背景磁场附近一个特殊的三维磁流体力学系统的整体适定性和稳定性。空间域为R^3，速度服从只有单向耗散的三维Navier-Stokes方程。如果没有Poincar\‘{e}型不等式，这个问题似乎是不可能的。通过发现实验观察到的稳定化效应的数学机制，并引入几种创新的技术来处理导数损失困难，我们能够约束Navier-Stokes非线性，并解决期望的全局适定性和稳定性问题。
The small data global well-posedness of the 3D incompressible Navier-Stokes equations in $\mathbb R^3$ with only one-directional dissipation remains an outstanding open problem. The dissipation in just one direction, say $\partial_1^2 u$ is simply insufficient in controlling the nonlinearity in the whole space $\mathbb R^3$. The beautiful work of Paicu and Zhang \cite{ZHANG1} solved the case when the spatial domain is bounded in the $x_1$-direction by observing a crucial Poincar\'{e} type inequality. Motivated by this Navier-Stokes open problem and by experimental observations on the stabilizing effects of background magnetic fields, this paper intends to understand the global well-posedness and stability of a special 3D magnetohydrodynamic (MHD) system near a background magnetic field. The spatial domain is $\mathbb R^3$ and the velocity in this MHD system obeys the 3D Navier-Stokes with only one-directional dissipation. With no Poincar\'{e} type inequality, this problem appears to be impossible. By discovering the mathematical mechanism of the experimentally observed stabilizing effect and introducing several innovative techniques to deal with the derivative loss difficulties, we are able to bound the Navier-Stokes nonlinearity and solve the desired global well-posedness and stability problem.