To uncover that the magnetic field mechanism can stabilize electrically conducting turbulent fluids, we investigate the stability of a special two dimensional anisotropic MHD system with vertical dissipation in the horizontal velocity component and partial magnetic damping near a background magnetic field. Since the MHD system has only vertical dissipation in the horizontal velocity and vertical magnetic damping, the stability issue and large time behavior problem of the linearized magneto-hydrodynamic system is not trivial. By performing refined energy estimates on the linear system coupled with a careful analysis of the nonlinearities, the stability of a MHD-type system near a background magnetic field is justified for the initial data belonging to \(H^{3}(\mathbf{R}^{2})\) space. The authors also build the explicit decay rates of the linearized system.

为了揭示磁场机制可以稳定导电湍流流体，我们研究了特殊二维各向异性 MHD 系统的稳定性，该系统具有水平速度分量的垂直耗散和背景磁场附近的部分磁阻尼。由于MHD系统在水平速度和垂直磁阻尼方面仅具有垂直耗散，因此线性化磁流体动力系统的稳定性问题和大时间行为问题并非微不足道。通过对线性系统进行精细的能量估计，并仔细分析非线性，MHD 型系统在背景磁场附近的稳定性对于属于 \(H^{3}(\mathbf { R}^{2})\)空间。作者还建立了线性化系统的显式衰减率。