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: 0801120
• 负责人: Hans Lindblad
• 金额: $ 25万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-15 至 2012-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801120&HistoricalAwards=false
• 关键词: Existence; Solutions; Hyperbolic; Differential; Equations

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 are: (i) Do we have existence and uniqueness of solutions, and continuous dependence on data, in a certain class? (ii) Can solutions blow up? (e.g. black holes in general relativity) (iii) What is the long time behavior of solutions? More specifically, the PI is mainly working in two areas. One project is to study the problem of existence of global solutions of Einstein's equations and of other related equations. The first goal is to simplify, generalize and refine the existence results for Einstein's equations. A long term goal is to study the stability of large solutions like black holes in general relativity. This is related to one of the main problem is mathematical relativity, the cosmic censorship conjectures of Penrose. The question of stability or blowup of large solutions is also the main question now in nonlinear wave equations. Another project is studying 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. The first goal in this area is to prove local existence. 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. 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 to study hyperbolic differential equations.The PI and collaborators greatly simplified the existence proof for Einstein's equations and its generalizations and refinements will have a large impact. It makes it much easier for graduate students to get in to mathematical relativity. Moreover, the detailed asymptotic behavior they prove in harmonic coordinates will be useful for the physics community. The physicists are building large gravitational wave detectors to observe the Universe. In order to know what to look for there is a large effort in doing numerical calculations for Einstein's equations and the only successful attempts have been in harmonic coordinates. It is also conceivable that understanding the properties of and controlling the interface between two fluids could have industrial applications. In particular there is a version of the problem for plasma physics in magneto-hydrodynamics and controlling the plasma is needed for constructing fusion reactors.

PI的研究涉及数学物理中关于非线性双曲型微分方程组的基本数学问题。其中包括经典场论和连续介质力学中的许多重要方程，例如爱因斯坦的广义相对论方程和欧拉的流体方程。基本问题是：(I)在某一类中，我们是否存在解的存在唯一性以及对数据的连续依赖？(Ii)解决方案会失败吗？(例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，国际和平研究所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其他相关方程的整体解的存在性问题。第一个目标是简化、概括和提炼爱因斯坦方程的存在结果。一个长期的目标是研究像广义相对论中的黑洞这样的大解决方案的稳定性。这其中涉及到的一个主要问题是数学相对性，即彭罗斯的宇宙审查猜想。大解的稳定性或爆破问题也是目前非线性波动方程的主要问题。另一个项目是研究流体动力学和广义相对论中的一类问题，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。这一领域的第一个目标是证明当地的存在。一个长期目标是研究气态恒星等天体物理体的长期行为，以及其他流体和固体的界面问题。为了解决这些问题，PI和合作者正在开发新的技术，这些技术也可以用来研究许多其他问题。特别是，他们用几何方法研究双曲型微分方程。Pi及其合作者大大简化了爱因斯坦方程的存在性证明，它的推广和细化将产生很大的影响。这使得研究生更容易接触到数学相对论。此外，他们在调和坐标下证明的详细的渐近行为将对物理界有用。物理学家们正在建造大型引力波探测器来观测宇宙。为了知道要寻找的是什么，对爱因斯坦方程进行了大量的数值计算，唯一成功的尝试是在调和坐标下。了解两种流体的性质并控制两种流体之间的界面也是可以想象的，这可能具有工业应用。特别是，磁流体力学中的等离子体物理问题有一个版本，建造聚变反应堆需要控制等离子体。