Physical experiments and numerical simulations have demonstrated that background magnetic fields stabilize electrically conducting fluids. This paper establishes these observations as mathematically rigorous facts on a 2D magnetohydrodynamic (MHD) system. This system is anisotropic with the velocity equation involving only the vertical dissipation. Flows governed by the 2D Navier–Stokes equations with only vertical dissipation are not known to be stable. Under the influence of a background magnetic field, the velocity field is shown here to stabilize and decay in time through the coupling and the interaction. Mathematically we reduce the MHD system concerned here to a system of degenerate and damped wave equations and exploit the smoothing and stabilizing effects of the wave structure. We are able to prove that any perturbation near a background magnetic field remains asymptotically stable. In addition, certain explicit large time behavior is also established.

物理实验和数值模拟表明，背景磁场可以稳定导电流体。本文将这些观察结果确立为二维磁流体动力学 (MHD) 系统上数学上严格的事实。该系统是各向异性的，速度方程只涉及垂直耗散。由仅具有垂直耗散的 2D Navier-Stokes 方程控制的流动未知是稳定的。在背景磁场的影响下，此处显示的速度场通过耦合和相互作用随时间稳定和衰减。在数学上，我们将此处涉及的 MHD 系统简化为退化和阻尼波动方程系统，并利用波浪结构的平滑和稳定效应。我们能够证明背景磁场附近的任何扰动都保持渐近稳定。此外，还建立了某些显式的大时间行为。