Multi-Dimensional Problems For Systems Of Conservation Laws

流体自由边界问题出现在许多物理、医学和工程模型中。与固定边界不同，自由边界是由问题本身的动态决定的，一个典型的玩具例子是在一杯水中融化的一块冰的边界。在天体物理学或多相流中研究恒星边界的动力学时，会出现流体-真空界面问题。在多相流体(如血液)中，会出现流体-流体界面问题。它们经常涉及对流体-可变形结构界面的研究，例如细胞变形。物理真空是一种自然介质(或者更确切地说，没有任何介质)，流体可以在其中扩散，例如在气态恒星的边界运动中，或者在流经多孔介质的流动中气体或液体与真空之间的界面传播时。本研究项目研究流体力学和天体物理中一些多维守恒律系统的真空自由边界问题。本项目将发展一些新的解析和几何方法来实现以下目标：1)建立气态恒星三维空间Euler-Poisson方程和Navier-Stokes-Poisson方程的真空自由边界问题的长时间适定性理论和理解长时间动力学，捕捉物理真空奇性；2)建立气态恒星三维空间Navier-Stokes-Poisson方程真空自由边界问题的消失粘性极限理论，捕捉物理真空奇性；3)阐明热导率对气态恒星物理真空边界动力学的作用。4)建立三维含阻尼型可压缩欧拉方程真空自由边界问题解的长时间适定性理论，并理解与多孔介质方程Barenblatt自相似解有关的解的界面行为。本项目将发展的新思想和新技术将有助于建立退化双曲、双曲-抛物线和双曲-椭圆型自由边界问题的一般理论。该项目还将为一些研究生和本科生提供应用数学的培训机会。

Fluid free boundary problems arise in many physical, medical, and engineering models. In contrast with fixed boundaries, free boundaries are determined by the dynamics of the problem itself, a typical toy example being the boundary of a piece of ice melting in a glass of water. Problems of fluid-vacuum interfaces arise in the study of dynamics of boundaries of stars in astrophysics or in multi-phase flows. In multi-phase fluids (like blood) fluid-fluid interfaces problems arise. Frequently they involve the study of fluid-deformable structure interfaces such as in cell deformation. Physical vacuum is a natural medium (or rather the lack of any medium) where fluids may spread, such as in the boundary motion of gaseous stars, or propagation of the interface between the gas or liquid and vacuum in flows through porous media. This research project deals with vacuum free boundary problems for some systems of conservation laws in multi-dimensions arising in fluid dynamics and astrophysics. Some new analytic and geometric methods will be developed in this project to achieve the following goals: 1) to establish the long time well-posedness theory and understand long time dynamics for the vacuum free boundary problems for the three spatial dimensional Euler-Poisson and Navier-Stokes-Poisson equations of gaseous stars, capturing the physical vacuum singularity, 2) to establish the vanishing viscosity limit theory for the vacuum free boundary problems for the three spatial dimensional Navier-Stokes-Poisson equations of gaseous stars, capturing the physical vacuum singularity, 3) to elucidate the role of the heat conductivity to the dynamics of physical vacuum boundary of gaseous stars, 4) to establish the long time well-posedness theory for solutions to the vacuum free boundary problem for the three spatial dimensional compressible Euler equations with damping and understand the interface behavior of solutions related to the celebrated Barenblatt self-similar solutions to porous media equations. New ideas and techniques to be developed in this project will contribute to a general theory of degenerate hyperbolic, coupled hyperbolic-parabolic and hyperbolic-elliptic free boundary problems. This project will also provide training opportunities to some graduate and undergraduate students in applied mathematics.