Existence of Solutions to Hyperbolic Differential Equations in Mathematical Physics

PI的研究涉及数学物理中非线性双曲微分方程系统的基本数学问题。其中包括经典场论和连续介质力学中的许多重要方程；爱因斯坦的广义相对论方程和欧拉的流体方程。基本问题涉及解的存在性、唯一性和稳定性，以及解是否会爆炸的问题（如广义相对论中的黑洞），如果不会爆炸，解的长时间行为是什么？更具体地说，PI主要在两个领域开展工作。其中一个项目是研究爱因斯坦方程和相关方程是否在任何时候都有解，或者解是否会爆炸。长期目标是研究广义相对论中黑洞和大爆炸等大解的稳定性。其动机是了解我们宇宙的大尺度结构。物理学家正在建造大型引力波探测器来观测宇宙，理论必须与观测一起发展。另一个项目是研究流体动力学和广义相对论中出现的一类问题，特别是证明真空中流体表面（如海洋表面）运动的自由边界问题的存在性和稳定性。长期目标是研究天体物理体的长期行为，如气态恒星以及其他流体和固体的界面问题。可以想象，了解两种流体之间界面的性质和控制可能具有重要的应用。特别是在磁流体动力学中有一个版本的问题，控制等离子体是建造聚变反应堆所必需的。为了解决这些问题，PI和合作者正在开发新的技术，这些技术也可以用于研究许多其他问题。特别是用几何方法结合频率分解方法来研究双曲型微分方程。PI和合作者极大地简化了爱因斯坦方程及其推广和改进的存在性证明，这将产生巨大的影响。此外，它们在调和坐标系中证明的详细的渐近行为也将对物理界有用。PI为流体自由边界问题开发的方法也适用于可压缩情况和涡量情况，因为它使用内部方程而不仅仅是边界方程。PI和合作者正在开发的处理变系数非线性方程的方法也有望用于显示大解扰动的稳定性。

The PI's research concerns basic mathematical questions about systems of nonlinear hyperbolic differential equations in mathematical physics. These include many important equations in classical field theory and continuum mechanics; e.g. Einstein's equations of general relativity and Euler's equations of fluids. The basic questions concern existence, uniqueness and stability of solutions, as well as the question if solutions blow up (e.g. black holes in general relativity) and if not what the long time behavior of solutions are? More specifically, the PI is mainly working in two areas. One project is to study if Einstein's and related equations have solutions for all times or if the solutions blow up. A long term goal is to study the stability of large solutions like black holes and the big bang in general relativity. The motivation is to understand the large scale structure of our universe. The Physicists are building large gravitational wave detectors to observe the universe and the theory has to be developed together with observations. Another project is to study a class of problems that occur in fluid dynamics and general relativity, in particular, proving existence and stability for the free boundary problem of the motion of the surface of a fluid in vacuum (such as the surface of the ocean). A long term goal is to study the long time behavior of astrophysical bodies such as gaseous stars as well as other interface problems of fluids and solids. It is conceivable that understanding the properties of and controlling the interface between two fluids could have important applications. In particular there is a version of the problem in magneto-hydrodynamics and controlling the plasma is needed for constructing fusion reactors.To solve these problems the PI and collaborators are developing new techniques that could be useful for studying many other problems as well. In particular, they are using geometric methods combined with frequency decomposition methods to study hyperbolic differential equations. The PI's and collaborator's greatly simplified existence proof for Einstein's equations and its generalizations and refinements will have a large impact. Moreover the detailed asymptotic behavior they prove in harmonic coordinates will be useful also for the physics community. The methods the PI has developed for the free boundary problem of fluids work also for the compressible case and also with vorticity since it uses interior equations and not just equations on the boundary. The methods PI and collaborators are developing to deal with nonlinear equations with variable coefficients will hopefully also be useful to show stability of perturbations of large solutions.