Abstract The stability and large-time behavior problem on some partially dissipated systems is not well-understood. The vorticity gradient of the 2D incompressible Euler equation can grow double exponentially in time while the same quantity to the 2D Navier-Stokes equation decays algebraically in time. However, the stability and large-time behavior of the vorticity gradients of the 2D Navier-Stokes equation with only vertical or horizontal dissipation appears to be unknown. This paper presents a global stability result on perturbations near a background magnetic field to the 2D incompressible magnetohydrodynamic (MHD) equations with vertical dissipation and horizontal magnetic diffusion. This stability result provides a significant example for the stabilizing effects of the magnetic field on electrically conducting fluids. In addition, we obtain an explicit decay rate for the solution of this nonlinear system.

中文翻译：

---

具有混合部分耗散的二维不可压缩 MHD 方程的背景磁场附近扰动的稳定性

摘要 一些部分耗散系统的稳定性和大时间行为问题尚未得到很好的理解。二维不可压缩欧拉方程的涡量梯度可以随时间成双指数增长，而二维纳维-斯托克斯方程的相同量随时间代数衰减。然而，只有垂直或水平耗散的二维纳维-斯托克斯方程的涡度梯度的稳定性和长时间行为似乎是未知的。本文针对具有垂直耗散和水平磁扩散的 2D 不可压缩磁流体动力学 (MHD) 方程，提出了背景磁场附近扰动的全局稳定性结果。这个稳定性结果为磁场对导电流体的稳定作用提供了一个重要的例子。此外，