Existence of Solutions to Hyperbolic Differential Equations in Mathematical Physics

PI: Hans Lindblad，加州大学圣地亚哥分校goms -0200226摘要：Lindblad的研究涉及数学物理中非线性双曲方程系统的基本数学问题。这些包括经典场论和连续介质力学以及经典物理学中的几个重要方程，例如爱因斯坦广义相对论方程和流体和弹性体方程。这些基本问题是：(i)在某一类中解是否局部存在唯一性？（ii）我们是否有放大的解决方案？（如广义相对论中的黑洞）（iii）解的长时间行为是什么？更具体地说，林德布拉德主要从事两个项目。一是证明流体动力学和广义相对论中一类问题的适定性，特别是证明真空中流体表面运动的自由边界问题的适定性。长期目标是研究天体物理体的长期行为，如气态恒星和海洋表面。另一个项目是研究与爱因斯坦方程相关的方程的全局解。一个长期的目标是简化和推广爱因斯坦方程的整体存在性结果。这两个问题涉及到物理学中的基本方程是否有全局解的问题。这些问题的解决可能会产生重要的后果。例如，可以想象，人们可以利用从爱因斯坦方程的解中获得的知识来允许使用引力波来观察宇宙。了解和控制两种流体之间的界面的性质可能具有工业应用价值。为了解决这些问题，林德布拉德和他的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，他们用几何方法来研究双曲型微分方程。

PI: Hans Lindblad, Univ. of Cal. San DiegoDMS-0200226Abstract:Lindblad's research concerns basic mathematical questions for systems of nonlinear hyperbolic equations in mathematical physics. These include several important equations in classical field theory and continuum mechanics as well as in classical physics, e.g. Einstein's equations in general relativity and the equation of fluids and elastic bodies. These basic questions are: (i) Do we have local existence and uniqueness of solutions in a certain class? (ii) Do we have blow-up of solutions? (e.g. black holes in general relativity) (iii) What is the long time behavior of solutions? More specifically, Lindblad is mainly working on two projects. One on proving the well-posedness for a class of problems that occur in fluid dynamics and general relativity, in particular proving the well-posedness for the free boundary problem of the motion of the surface of a fluid in vacuum. A long term goal is to study the long time behavior of astrophysical bodies such as gaseous stars as well as the surface of the ocean. Another project is to study global solutions of equations related to Einstein's equations. A long term goal is to simplify and generalize the global existence results for Einstein's equations. These two problems are related to the question of whether the fundamental equations in physics have global solutions. Solution to these questions could have important consequences. For instance, it is conceivable that one could use the knowledge obtained from the solutions of Einstein's equations to permit the use of gravitational waves to observe the Universe. Understanding the properties of and controlling the interface between two fluids could have industrial applications. To solve these problems Lindblad and his collaborators are developing new techniques that could be useful for studying many other problems as well. In particular, they are using geometric methods to study hyperbolic differential equations.