Existence of Solutions to Hyperbolic Differential Equations in Mathematical Physics

数学物理中双曲微分方程解的存在性

• 批准号: 0200226
• 负责人: Hans Lindblad
• 金额: $ 11.13万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2006-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200226&HistoricalAwards=false
• 关键词: Existence; Solutions; Hyperbolic; Differential; Equations

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.

主要研究者：Hans Lindblad，加州大学圣地亚哥分校DMS-0200226摘要：Lindblad的研究涉及数学物理中的非线性双曲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以及经典物理学中的几个重要方程，例如广义相对论中的爱因斯坦方程和流体和弹性体方程。 这些基本问题是：（i）解的局部存在唯一性 在某个班级？ (ii)我们有解的爆破吗？ (e.g.（3）解的长时间行为是什么？更具体地说，Lindblad主要从事两个项目。一类是证明流体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。长期目标是研究气态恒星以及海洋表面等天体物理体的长期行为。另一个项目是研究与爱因斯坦方程相关的方程的整体解。长期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整体解的问题。 这些问题的解决可能会产生重要的后果。例如，可以想象，人们可以使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。了解两种流体的性质并控制它们之间的界面可能具有工业应用价值。 为了解决这些问题，Lindblad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，他们正在使用几何方法研究双曲微分方程。