流体力学方程组中若干边值问题的研究

具有边界的流体力学方程组，尤其是自由边值问题和无粘极限问题，是流体力学中一类重要模型，广泛应用于海洋运动，大气物理，天体物理，航天航空等领域。该领域仍有众多公开问题待解决。.本项目的研究课题如下：浅滩问题的适定性；可压缩浅滩问题的适定性；混合两相流的适定性；薄板区域上的无粘极限。本项目拟通过深挖浅滩问题的结构，推导与角点奇性相匹配的权重从而证明适定性；深入研究可压缩Euler方程与不可压缩Euler方程的区别和联系，明确散度在自由边界问题中影响从而建立可压缩浅滩问题的研究框架；在表面张力的稳定性作用下，通过建立统一框架研究可压缩Euler方程与不可压缩Euler方程，从而得到混合两相流的适定性；通过高阶展开、边界提升等方法，研究薄板区域上的无粘极限问题。.本项目的研究将有助于进一步完善和发展边界问题的理论和方法，为探索天体物理奥秘、物理实验及应用提供数学理论指导。

The fluid dynamical system in a domain with boundary is a very important model in mathematical fluid mechanics, especially the free boundary problems and inviscid limit problems. It has a wide range of applications in oceanography, atmosphere physics, astrophysics, aviation and other theoretical and engineering aspects. By now, there are still many open problems..The project plan to study the following problems: well-posedness of beach problem; well-posedness of the compressible beach problem; well-posedness of the mixed two-phase fluid; the inviscid limit in a thin domain. To do that, we will deeply analyze the structure of the beach problem to get the suitable weight which is according to singularity of the corner。We will analyze the structure of the equations to get the relationship between the compressible Euler equations and incompressible Euler equations and the effect of the divergence part in the compressible Euler equations. Then we establish a framework to study the compressible beach problems. Based on the surface tension, we will establish a uniform framework to study the compressible Euler equations and incompressible Euler equations to get the well-posedness of mixed two-phase fluid. Based on high-order expansions, boundary layer lift and other methods, we will study the inviscid limit in a thin domain..This project will help to further improve the theory in the field of the problems with boundary and provide mathematical theoretical guidance for exploring the mysteries of astrophysics, physical experiments and applications.