可压缩流体及相关方程组中的真空自由界面问题

在此项目中，我们将研究可压缩流体力学`和广义相对论流体偏微分方程组的流体和真空的自由界面问题。其主要目的是为高维退化双曲型，退化双曲-抛物型，退化双曲-椭圆型方程组的自由边值问题发展一些新的分析和几何方法以实现以下目的：1）刻画真空边界的物理奇性，2）阐明热传导，引力，磁场等对真空自由边值问题的适定性和解的定性行为的影响， 3）建立广义相对论流体偏微分方程组真空自由边值问题的球对称解的适定性理论。.流体自由边值问题来源于许多物理，医学及工程学中的模型，例如，星体的形状，多相流，反应流，生物医学中的细胞形变等。关于流体自由边值问题的研究一直是极具挑战和热门的课题。在本项目中发展的分析和几何方法将会在物理，化学，生物，工程中的相关问题的研究起作用， 并高维退化双曲型，退化双曲-抛物型，退化双曲-椭圆型方程组的自由边值问题的一般理论有所贡献。

This research project deals with free boundary problems with the interfaces separating the fluids and vacuum for some systems of PDEs arising in fluid dynamics and general relativistic fluids. The main purpose is to develop some new analytic and geometric methods for free boundary problems of multi-dimensional degenerate hyperbolic, hyperbolic-parabolic and hyperbolic-elliptic equations 1) to capture the physical singularity at the vacuum boundary, 2) to elucidate the role of the heat-conductivity, gravitation and magnetic fields to the well-posedness and qualitative behavior of solutions for the vacuum free boundary problems; 3) to establish the well-posedness for the vacuum free boundary problem for spherical symmetric solutions of general relativistic fluids.. Fluid free boundary problems arise in many physical, medical, and engineering models, for example, shape of stars, multi-phase flow, reacting flow, biomedical modeling such as cell deformation and etc. The study of free boundary problems in fluids has been a very challenging and hot topic. The analytic and geometric ideas and methods to be developed in this project will be useful to other related problems arising in physics, chemistry, biology and engineering.. and will contribute to a general theory of degenerate hyperbolic, coupled hyperbolic-parabolic and hyperbolic-elliptic free boundary problems.