不可压缩磁流体力学方程弱解的研究

This program is dedicated to studying the regularity of the (suitable) weak solutions of the 3D MHD equations by utilizing the tools such as Blow-up analysis, Backward heat kernel, Backward uniqueness and Unique continuation of heat operators, the trajectories method and the classical elliptic and parabolic equations theory. Particularly:.(1) We intend to give the limiting case of Serrin-type regularity criteria via the velocity field only for the weak solutions of the tri-dimensional MHD equations..(2) We want to present some higher derivatives estimate of the weak solutions for the 3D MHD equations..(3) We plan to establish the regularity criterion just in terms of the gradient of the velocity field or the vorticity of the velocity to the suitable weak solutions for the 3D MHD equations..(4) We want to obtain the partial regularity of the suitable weak solutions to the three-dimensional Hall-MHD and derive higher derivatives estimate of weak solutions for this system. .The investigation of this project will enrich the investigation of the regularity of the (suitable) weak solutions of the 3D MHD and Hall-MHD equations and provides the theoretical support for the application of MHD equations.

不可压缩磁流体方程广泛应用于地球物理，天体力学，等离子体物理的研究当中。本项目拟采用经典的椭圆抛物方程理论、Blow-up分析方法、倒向热核、热算子倒向唯一性、热算子唯一性延拓方法、粒子轨道技术等工具，围绕以下几个问题开展研究：.（1）对三维不可压缩磁流体MHD方程的弱解，建立只依赖速度场的Serrin型端点正则性准则。.（2）推导三维不可压缩磁流体MHD方程弱解在Lorentz 空间的高阶导数估计。.（3）研究三维不可压缩磁流体MHD方程适当弱解的正则性问题，给出适当弱解只需速度场梯度（旋度）的正则性准则。.（4）探究三维Hall-MHD方程适当弱解的部分正则性并给出其弱解的高阶导数估计。.通过本项目的实施，增加对磁流体力学方程的数学认识，展现磁场和速度场的相互作用，丰富和完善磁流体力学方程弱解的正则性结果，为其应用提供理论支撑。

不可压缩磁流体力学方程组和其子系统Navier-Stokes方程在等离子体和电解质等导电流体中起着至关重要的作用。本项目采用经典的椭圆抛物方程理论、Blow-up分析方法、倒向热核、热算子倒向唯一性、热算子唯一性延拓方法、粒子轨道技术等工具，围绕以下三个问题进行了持续深入研究并取得了系列进展：. (1)三维磁流体力学方程组及其子系统Navier-Stokes方程组适当弱解可能奇异点集的上界Box维数问题。. (2)三维磁流体力学方程组及其子系统Navier-Stokes方程组适当弱解在单个尺度和多个尺度上的正则性。. (3)三维磁流体力学方程组及其子系统Navier-Stokes方程组弱解在Lorenz空间中的正则性。.通过本项目的实施，我们对磁流体力学方程组及其子系统Navier-Stokes方程组的结构和性质有了更全面的了解和更深层次的认识，同时也提高了我们的数学研究水平。