Existence and Blow-Up of Solutions to Systems of Nonlinear Wave Equations

Lindblad的研究涉及非线性波动方程系统的基本数学问题，包括物理学中的几个重要方程，例如经典场论和连续介质力学，以及经典物理学中的爱因斯坦方程，杨-米尔斯方程和非线性弹性方程。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，林德布拉德正在从事两个项目。一个是和Demetri Christodoulou一起证明了流体动力学和广义相对论中一类问题的适定性，特别是证明了真空中流体表面的自由边界问题的适定性。另一个项目是研究如何规则的初始数据，以确保非线性波动方程的局部解的存在，特别是构造局部存在的反例。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。为了解决这些问题，Linblad和他的合作者正在开发全新的技术，这些技术也可以用于研究许多其他问题，例如两种流体之间界面的性质。这样的结果将具有工业应用价值。

Lindblad's research concerns basic mathematical questions for systems of nonlinear wave equations which include several important equations in physics, for instance, in classical field theory and continuum mechanics as well as in classical physics, e.g. Einstein's equations, the Yang-Mills equation and the equation of nonlinear elasticity. 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 working on two projects. One with Demetri Christodoulou on proving the well-posedness for a class of problems that occurr in fluid dynamics and general relativity, in particular proving the well-posedness for the free boundary problem of the surface of a fluid in vacuum. Another project is to study how regular initial data needed to ensure the existence of a local solution to nonlinear wave equations, in particular constructing counterexamples to local existence. 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. To solve these problems Linblad and his collaborators are developing completely new techniques that could be useful for studying many other problems as well, e.g. the properties of the interface between two fluids. Such a result would have industrial applications.