磁流体力学方程组的稳定性和不稳定性研究

磁流体力学方程组的稳定性和不稳定性是近年来偏微分方程领域研究的热点和难点问题之一。本项目主要围绕磁流体力学方程组的整体适定性、非均匀稳态磁场的稳定性与不稳定性、Rayleigh-Taylor不稳定性等问题展开，重点研究无磁耗散含真空可压缩磁流体力学方程组大初值整体适定性以及该方程组自由边值问题非均匀稳态磁场水平方向为零，垂直方向不均匀的稳定性和垂直方向为零，水平方向不均匀的不稳定性；研究含真空可压缩磁流体力学方程组自由边值问题在非均匀稳态磁场扰动下的Rayleigh-Taylor不稳定性；研究无磁耗散含真空可压缩磁流体力学方程组自由边值问题在柱对称稳态磁场扰动下的稳定性和不稳定性；研究具有磁耗散的磁流体力学方程组自由边值问题当粘性系数和磁耗散系数趋于零时的极限问题以及相关的初始层、边界层理论。所研究内容均来源于实际物理问题，具有很强的应用背景，在数学上具有重要理论意义。

Stability and instability of the magneto-hydrodynamic (MHD) equations are one of the hot and difficult issues in the field of PDEs in recent years. This project focuses on global well-posedness, stability and instability of inhomogeneous equilibrium magnetic field, Rayleigh-Taylor instability. Including global well-posedness for the non-resistive compressible MHD equations with vacuum and large initial data, stability of inhomogeneous equilibrium magnetic field where horizontal component is zero and vertical component is variant and instability of inhomogeneous equilibrium magnetic field where vertical component is zero and horizontal component is variant for the non-resistive compressible MHD equations with free boundary and vacuum, Rayleigh-Taylor instability under perturbation of inhomogeneous equilibrium magnetic field for the compressible MHD equations with free boundary and vacuum, stability and instability of axisymmetric equilibrium magnetic field for the non-resistive compressible MHD equations with free boundary and vacuum, and limit problems for the resistive compressible MHD equations with free boundary, then the associated analysis of the resistive compressible MHD equations with initial and boundary layers for these limit problems. These problems originate from practical problems with strong physical background and introduce many essential mathematical theoretical problems.