We consider stability problems for the compressible viscous and diffusive magnetohydrodynamic (MHD) equations arising in the modeling of magnetic field confinement nuclear fusion. In the first part, we investigate the Cauchy problem to the barotropic MHD system. With the help of the techniques of anti-symmetric matrix and an induction argument on the order of the space derivatives of solutions in energy estimates, we prove that smooth solutions exist globally in time near the non-constant equilibrium solutions. We also obtain the asymptotic behavior of solutions when the time goes to infinity. The result shows that gradients of both the velocity and the magnetic field converge to the equilibrium solutions with the same norm \( \Vert \cdot \Vert _{H^{s-3}}\), while the density converge with stronger norm \( \Vert \cdot \Vert _{H^{s-1}}\). In the second part, the initial value problem to the full MHD system is studied. By means of the techniques of choosing a non-diagonal symmetrizer and elaborate energy estimates, we prove the existence and uniqueness of global solutions to the system when the initial data are close to the non-constant equilibrium states. We find that both the density and temperature converge to the equilibrium states with the same norm \( \Vert \cdot \Vert _{H^{s-1}}\). These phenomena on the charge transport show the essential relationship of the equations between the barotropic and the full MHD systems.

我们考虑在磁场约束核聚变建模中出现的可压缩粘性和扩散磁流体动力学（MHD）方程的稳定性问题。在第一部分中，我们研究了正压MHD系统的柯西问题。借助反对称矩阵技术和能量估计中解的空间导数阶数的归纳论证，我们证明了光滑解在时间上全局存在于非恒定平衡解附近。当时间到无穷远时，我们还获得了解的渐近行为。结果表明，速度和磁场的梯度都收敛到具有相同范数\（\ Vert \ cdot \ Vert _ {H ^ {s-3}} \）的平衡解，而密度收敛于更强的范数\（\ Vert \ cdot \ Vert _ {H ^ {s-1}} \）。在第二部分中，研究了完整MHD系统的初始值问题。通过选择非对角对称器的技术和详尽的能量估计，我们证明了当初始数据接近非恒定平衡状态时系统整体解的存在性和唯一性。我们发现密度和温度都以相同的标准\（\ Vert \ cdot \ Vert _ {H ^ {s-1}} \）收敛到平衡状态。电荷传输中的这些现象表明了正压和完整MHD系统之间方程的本质关系。