Nonlinear Wave Equations

本文的目的是研究非线性波动方程低正则解的性质。这个问题现在对于一般方程已经很好理解了。然而，物理学中产生的大多数非线性波动方程（例如，广义相对论中的爱因斯坦方程（Einstein's equations in general relativity）表现出某种特殊的结构，人们期望这种结构保证解的行为比一般方程好得多。正是这类方程是拟议中的研究重点。这一主题在很大程度上尚未探索，直到现在，被认为是遥不可及的。然而，PI最近的工作可能是这一领域取得进展的关键。

The aim of the proposed work is to study the behavior of low regularity solutions to nonlinear wave equations. This problem is by now well understood for generic equations. However, most nonlinear wave equations arising in physics (e.g., Einstein's equations in general relativity) exhibit some kind of special structure which, one expects, guarantees a much better behavior of solutions than for a generic equation. It is precisely this type of equations that the proposed research will focus on. This subject is largely unexplored, and, until now, was considered to be out of reach. However, recent work of the PI may hold the key to progress in this field.