磁流体力学方程组的稳定性和不稳定性研究

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.

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。

本项目主要研究了流体力学和等离子体物理中的磁流体力学方程组及相关模型的数学理论问题，特别关注这些模型的稳定性和不稳定性以及相关的边界层理论等。经过四年的研究，本项目主要建立了：（1）平面无磁阻可压缩磁流体力学方程组柯西问题大初值强解的整体存在性；（2）高维粘性和没有粘性的含真空可压缩磁流体力学方程组自由边值问题的Z-pinch线性不稳定性；（3）具有温度和部分耗散的磁流体力学方程组在一致磁场作用下Couette流和动力学平衡解的稳定性；（4）对于Euler稳定的profile情形，三阶项具有抑制Prandtl边界层的线性不稳定性的作用；（5）三维情形时，无热耗散具有磁耗散并且动量方程上具有阻尼项的磁流体力学方程组大初值整体光滑解的存在唯一性，以及无热耗散和无磁耗散情形该方程组大初值轴对称整体光滑解的存在唯一性；（6）具有物理边界的非等熵可压缩Navier-Stokes方程组在半空间或者球外有限时间爆破；（7）具全部粘性或者水平粘性本原方程的局部或者整体适定性理论；（8）当宽长比趋于零时，对应的伸缩Navier-Stokes方程的强解和z-弱解分别强收敛到本原方程的强解和z-弱解；（9）二维稳态磁流体力学方程组在水平磁场与水平流场之比远远小于1及其它相应假设下，对于外流为剪切流和非剪切流的情况下，边界层展开的有效性。这些结果的取得较好地完成了项目申请书提出的问题，必将为后续的深入研究和流体力学中千禧年大奖难题的解决提供必要的前期研究基础。这些结果目前已经在Communications on Pure and Applied Mathematics、Advances in Mathematics、Mathematical Models and Methods in Applied Sciences、SIAM Journal on Mathematical Analysis、Archive for Rational Mechanics and Analysis、Journal of Mathematical Fluid Mechanics、Journal of Differential Equations、Journal of Mathematical Physics、Science China Mathematics等主流学术期刊上公开发表。