关于流体运动方程组能量守恒的若干数学问题

流体运动方程组能量守恒问题是近年来国际应用数学界的重要研究课题之一。本项目将研究两类具有实际应用背景的流体运动方程组能量守恒问题。第一类是非牛顿流体力学方程组，其剪切应力与剪切速率之间是非线性关系，描述了流体的流变性质。第二类是非线性粘性介质的磁流体力学方程组，描述了非线性粘性介质在磁场作用下的运动规律。由于上述两类流体运动方程组具有复杂的非线性结构，对它们的研究是一项具有挑战性的工作。我们拟采用全局磨光技巧、正则性分析方法、交换子估计等技巧和方法，通过巧妙利用方程组的结构来探索非牛顿流体力学方程组与非线性粘性介质的磁流体力学方程组的能量守恒问题，有望得到一些新的结果。这将会丰富和完善流体运动方程组的数学理论。

Energy conservation problem in fluid mechanics is one of the most important topics in applied mathematics. This project will study energy conservation for weak solutions of two type of partial differential equations which have practical application background. One is the dynamic model of non-Newtonian fluids, which have nonlinear relation between stress tensor and velocity tensor, describe the rheological properties of fluids. The other is the model of magnetohydrodynamic equations for nonlinearly viscous media, which describe the motion of nonlinearly viscous media in magnetic field. As the strong nonlinearity of the above two type dynamic equations, our research is a challenging work. Except for the global mollification methods, regularization analysis methods, commutator estimates, we also need to use the structure of the equations to explore the energy conservation of non-Newtonian fluids and the modified magnetohydrodynamic equations. We expect to get some new results, which shall rich and perfect the mathematical theory in fluid mechanics.