论文标题
全球解决3D不可压缩的MHD系统，仅在一个方向上耗散
Global solutions to 3D incompressible MHD system with dissipation in only one direction
论文作者
论文摘要
3D不可压缩的Navier-Stokes方程在$ \ Mathbb r^3 $中，只有一个方向耗散仍然是一个悬而未决的开放问题。仅在一个方向上的耗散，例如$ \ partial_1^2 u $根本不足以控制整个空间中的非线性$ \ Mathbb r^3 $。 Paicu和Zhang \ cite {Zhang1}的精美作品解决了空间域通过观察至关重要的Poincaré型不平等而以$ x_1 $方向界定的情况。本文旨在了解背景磁场的稳定作用的实验性观察，该论文旨在了解背景磁场附近的特殊3D磁性水力动力学（MHD）系统的全球稳定度和稳定性。空间域是$ \ mathbb r^3 $，此MHD系统中的速度仅具有一个方向耗散的3D Navier-Stokes。由于没有庞加莱的类型不平等，这个问题似乎是不可能的。通过发现实验观察到的稳定效果的数学机制，并引入了几种创新技术来应对衍生性损失困难，我们能够束缚Navier-Stokes非线性并解决所需的全球全球良好性和稳定性问题。
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 Poincaré 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 Poincaré 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.
