一类二维非线性守恒律方程组黎曼解结构的研究

Both the triple-shock pattern and the Mach-reflection-like configuration are important structures of solutions to the multi-dimensional systems of nonlinear conservation laws. In this project, we study the Riemann problem for a two-dimensional system of nonlinear conservation laws which models polymer flooding in oil reservoir. Employing the theories of both multi-dimensional system of nonlinear conservation laws and generalized functions, with the generalized characteristic analysis method and the numerical simulations, we analyze the interactions among the two-dimensional elementary waves completely. Besides, we will establish both the conditions and the mathematical mechanisms for the appearance of the triple-shock pattern as well as the Mach-reflection-like configuration. Furthermore, the conditions for the appearance of the Mach-reflection-like configuration caused by the local and global interactions among the two-dimensional elementary waves will be also proposed. Finally, the corresponding Riemann problems will be solved. The studies of this project are very important to establish the mathematical theories about the multi-dimensional systems of nonlinear conservation laws.

三激波解结构、类马赫反射解结构都是高维非线性守恒律方程组中重要的解结构。本项目研究在石油井中描述聚合物驱油过程的一个二维非线性守恒律方程组的黎曼问题。拟运用高维非线性守恒律方程理论，广义函数理论，广义特征分析方法，数值模拟等理论和研究方法，深入地分析各二维初等波的相互作用。建立三激波解结构、类马赫反射解结构出现的条件和数学机理。提出由二维初等波的局部和整体相互作用所产生的类马赫反射解结构出现的条件，并解决相应的黎曼问题。该项目的研究将有助于建立高维非线性守恒律方程组的数学理论。

在本项目中，我们对非线性守恒律方程组的黎曼解结构进行了研究。在一类二维非线性守恒律方程组的黎曼问题研究中，分析了各二维初等波的相互作用，发现了由二维初等波的局部和整体相互作用所产生的类马赫反射结构。此外，对Chaplygin气体动力学方程组的黎曼问题进行了研究。发现了在多个状态变量上都出现狄拉克函数的狄拉克激波，并建立了该狄拉克激波的显式表达式。获得了包含类马赫反射结构、狄拉克激波等一些具有不同几何结构的黎曼解。发表了4篇论文,较好地完成了本项目的研究工作。