Existence of Solutions to Hyperbolic Differential Equations in Mathematical Physics

数学物理项目中双曲型微分方程解的存在性本项目将集中研究某些非线性双曲型偏微分方程组(PDE)的适定性。一类问题是真空中流体表面运动的自由边界问题。另一个项目是研究与爱因斯坦广义相对论方程有关的方程的整体解。这些方程的基本问题包括：(I)我们在某一类中是否存在解的局部存在性和唯一性？(Ii)我们有解决方案的爆破吗？(例如广义相对论中的黑洞)(Iii)解的长时间行为是什么？这些问题的答案可能会提供对宇宙中引力波的进一步理解。了解两种流体的性质并控制两种流体之间的界面可能具有工业应用。

Existence of Solutions to Hyperbolic Differential Equations in Mathematical PhysicsProject AbstractHans LindbladThis project will concentrate on studying the well-posedness of certain nonlinear systems of hyperbolic partial differential equations (PDEs). One class of problems is the free boundary problem for the motion of the surface of a fluid in a vacuum. Another project is to study global solutions of equations related to Einstein's equations of general relativity. The basic questions for these equations include: (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?The answers to these questions may provide further understanding of gravitational wavesin the universe. Understanding the properties of, and controlling the interface between, two fluids may have industrial applications.