论文标题
3D不可压缩的MHD方程的稳定性和大型行为，背景磁场附近有部分耗散
Stability and large-time behavior on 3D incompressible MHD equations with partial dissipation near a background magnetic field
论文作者
论文摘要
物理实验和数值模拟观察到了一个显着的稳定现象：背景磁场稳定，潮湿电动传导流体。本文旨在将这种现象确定为在磁性水力动力学（MHD）系统上具有各向异性耗散的数学严格事实，$ \ Mathbb r^3 $。该系统中的速度方程是仅在$ x_1 $方向上耗散的3D Navier-Stokes方程，而磁场则在两个水平方向上使用磁扩散来遵守感应方程。我们确定背景磁场附近的任何扰动$（0,1,0）$在Sobolev设置$ H^3（\ Mathbb r^3）$中均在全球范围内稳定。此外，还获得了$ h^2（\ Mathbb r^3）$中的显式衰减率。当没有磁场的存在时，$ \ Mathbb r^3 $中的3D各向异性Navier-Stokes方程不太了解，而小型数据全球范围良好仍然是一个有趣的开放问题。本文揭示了磁场如何产生增强耗散并有助于稳定流体的机制。
Physical experiments and numerical simulations have observed a remarkable stabilizing phenomenon: a background magnetic field stabilizes and damps electrically conducting fluids. This paper intends to establish this phenomenon as a mathematically rigorous fact on a magnetohydrodynamic (MHD) system with anisotropic dissipation in $\mathbb R^3$. The velocity equation in this system is the 3D Navier-Stokes equation with dissipation only in the $x_1$-direction while the magnetic field obeys the induction equation with magnetic diffusion in two horizontal directions. We establish that any perturbation near the background magnetic field $(0,1,0)$ is globally stable in the Sobolev setting $H^3(\mathbb R^3)$. In addition, explicit decay rates in $H^2(\mathbb R^3)$ are also obtained. When there is no presence of the magnetic field, the 3D anisotropic Navier-Stokes equation in $\mathbb R^3$ is not well understood and the small data global well-posedness remains an intriguing open problem. This paper reveals the mechanism of how the magnetic field generates enhanced dissipation and helps stabilize the fluid.
