坐标显含极角柱坐标系下MHD方程组整体光滑解的存在性及其对LL方程的应用

MHD方程包含7个数学千禧年问题之一不可压NS方程，而三维不可压Navier-Stokes方程整体光滑解的存在性是一个重大的公开问题，即使是在坐标不显含极角的柱坐标系下除部分初边值以外其整体光滑解的存在性仍是公开问题，由于坐标显含极角柱坐标系情形比不显含极角情形复杂得多，以致于关于坐标显含极角柱坐标系情形三维NS方程整体光滑解存在性的研究是空白. 我们在研究柱坐标系下三维MHD方程整体光滑解存在性时发现坐标显含极角有时恰恰能克服不显含极角时产生的能量估计的困难，本项目根据这一发现对前述空白情形（坐标显含极角柱坐标系下）研究可压或不可压三维MHD（含NS）方程整体光滑解的存在性，并尝试把它用于三维液晶方程和LL方程.

The MHD equation contains the incompressible NS equation which is one of the seven mathematical Millennium problems,. The existence of global smooth solutions for the three-dimensional incompressible Navier-Stokes equation is a major open problem. Even in cylindrical coordinates without polar angle, the existence of global smooth solutions is still an open problem except for some initial and boundary values. Because the coordinates with polar angle are more complex than those without polar angle, the existence of global smooth solutions for the three-dimensional incompressible Navier-Stokes equation is still an open problem. The situation is so complicated that the research on the existence of global smooth solutions of three-dimensional NS equation with polar coordinates is blank. When we study the existence of global smooth solutions of three-dimensional MHD equation in cylindrical coordinates, we find that sometimes the difficulty of energy estimation can be overcome when polar coordinates with polar angles are absent. The existence of globally smooth solutions for compressible or incompressible three-dimensional MHD (including NS) equations is studied in the coordinate explicit polar cylindrical coordinate system, and it is attempted to be applied to three-dimensional liquid crystal equations and LL equations.