退化抛物型方程的熵解适定性研究

This research is mainly on the well-posedness of entropy solutions to a strongly degenerate parabolic equation. Two problem is investigated: (1) Cauchy Problem; (2) Dirichlet problem with nonhomogeneous boundary. . The well-posedness of degenerate parabolic equation is one of the foci of nonlinear partial differential equation. It can be widely used in many fields to model lots of important problems, such as Reaction-diffusion model of biological population, the model of heat transfer in porous medium. For the strongly degenerate parabolic equation, the parabolic domain is tightly coupled with hyperbolic domain. It is well known that the solution of the hyperbolic equation is usually discontinuous, so the methods of parabolic equations are usually invalid. Fortunately, the theory of Kruzkov entropy solution open a filed for the research on the hyperbolic conservation law and degenerate parabolic equation.. The parabolic equation in this research is strongly degenerate, and includes various different degenerate partial differential equations such as hyperbolic conservation law, elliptic-parabolic equation, parabolic-hyperbolic equation. Since the equation has a more general form, it is important and practical to research on its well-posedness.

本项目研究一类强退化的抛物型方程，主要内容如下：（1）Cauchy问题的熵解适定性；（2）带有非齐次边界条件的Dirichlet问题的熵解适定性。. 退化抛物型方程的适定性研究是非线性偏微分方程中的一个热点问题，具有广泛的应用背景。对于强退化的抛物型方程，由于其抛物区域和双曲区域耦合在一起，且双曲方程的解往往会发生间断，故人们无法利用抛物方程的方法对其解的适定性进行研究。Kruzkov熵解的概念以及研究理论，为一阶双曲守恒律方程以及退化抛物型方程的研究开辟了新的天地。. 本项目研究的抛物型方程具有强退化性，可以涵盖多类退化偏微分方程，如：一阶双曲守恒律方程、椭圆-抛物型方程、抛物-双曲型方程等。由于该类方程具有更一般的形式，因此其熵解的适定性研究具有重要十分重要的研究价值与应用价值。