Navier-Stokes 和 MHD 方程的适定性问题研究

Navier-Stokes and MHD equations are important nonlinear differential equations in fluid dynamics as well as basic equations describing the motion of fluid in macro sense for our project.Its mathematical study is one of the frontiers in the area of nonlinear differential equations. Its contents include: outflow(inflow) problem of Navier-Stokes equations, the existence and long time behavior of Navier-Stokes equations and MHD equations with temperature dependent transport coefficients, time-periodic solution to Navier-Stokes equations for higher dimension and the partial regularity of suitable weak solutions for fractional Navier-Stokes equaitons and (hall) MHD equations. These problems are challenging and important in theoretical research. It needs to develop new idea and method since they are difficult and hot points in fluid dynamic equations. It also finds wide application since there is strong physical background and sense for Navier-Stokes and MHD equations.

本项目研究的Navier-Stokes和MHD方程是流体动力学中重要的非线性偏微分方程，亦是宏观意义下描述流体运动的基本方程。其数学理论研究一直是非线性偏微分方程研究的前沿问题之一。研究内容包含：Navier-Stokes方程的内（外）流问题、传输系数依赖温度的Navier-Stokes方程和MHD方程的存在性和大时间行为、高维Navier-Stokes方程的时间周期解和分数阶 Navier-Stokes 和（Hall）MHD方程弱解部分正则性。这些问题是流体力学方程研究的热点和难点之一，有重要的理论研究意义，极具挑战性，需要发展新思想和新方法。同时Navier-Stokes和MHD方程有强大的物理背景和意义，具有广泛的应用前景。

本项目在国家自然科学基金的资助, 项目组成员按照研究计划依次开展研究工作,项目进展顺利。我们主要研究了可压缩Navier-Stokes方程不稳定性、平面MHD方程的大时间行为和边界层和粘性消失的极限等方面的问题，其中平面MHD方程Dirchlet问题和Cauchy问题的的大时间行为是首次得到，我们证明了温差可以决定流体的稳定性，解决了大家长期关心的问题。