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

林德布拉德的研究涉及基本的数学问题系统的非线性波动方程，其中包括几个重要的方程在物理学，例如，在经典场论和连续介质力学以及在经典物理学，例如爱因斯坦方程，杨米尔斯方程和方程的非线性弹性。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正在进行两个项目。 一个是与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自由边界问题的适定性。另一个项目是研究如何需要规则的初始数据来确保非线性波动方程局部解的存在性，特别是构造局部存在性的反例。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。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，例如两种流体之间的界面特性。这样的结果将具有工业应用。

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.