时空流形中的非线性波动方程

The principle investigator's research area is in harmonic analysis and PDEs, particularly the theoretical investigation of the nonlinear wave equations. He is specialized in the study of the Strichartz estimates and the Strauss conjecture on various space-time manifolds. Among others, he and his coauthors proved radial endpoint Strichartz estimates (which was a conjecture of Klainerman in 1995) and gave a counterexample for the two dimensional endpoint Strichartz estimate; they proved weighted Strichartz estimates with angular regularity, with application to solve the Strauss conjecture with low regularity for spatial dimension two, three and four; they developed the generalized Strichartz estimates and applied it to the two dimensional Strauss conjecture on exterior domain; they also proved low dimensional Strauss conjecture on asymptotically Euclidean manifolds, and the radial Glassey conjecture. In this project, the principle investigator will concentrate on the study of the influence of the regularity and size of the initial data and the background manifolds on the long time behavior of the solutions for the nonlinear wave equations. The main topics of research will include the Strauss conjecture, the Glassey conjecture and quasilinear wave equations on spacetime manifolds like exterior domains, asymptotically flat manifolds, Schwarzschild spacetime and Kerr spacetime. In addition, he will also study the local wellposedness with low regularity and long time wellposedness with small data for nonlinear wave equations.

申请人主要从事非线性波动方程的理论研究，特别对于Strichartz估计以及各类时空流形上的Strauss猜想有较深的造诣。其中，我们证明了径向端点Strichartz估计(Klainerman猜想95），给出了二维端点估计的反例；证明了用角变量来刻画的加权Strichartz 估计，用于解决低维低正则Strauss猜想；引入广义Strichartz估计，解决二维外区域上的Strauss猜想；证明了径向Glassey猜想和低维渐近欧氏空间上的Strauss猜想。本项目中，我们将继续从事该领域的研究，瞄准有重大研究意义的问题，着重探讨初值正则性和尺度以及背景流形的变化对于非线性波动方程解的长时间性态的影响。主要的研究课题包括外区域、渐近平坦时空、Schwarzschild时空、Kerr时空等时空流形上的Strauss猜想，Glassey猜想和拟线性波动方程，以及非线性波动方程的低正则适定性。

本项目按计划顺利完成，完成SCI收录学术论文7篇，发表于CMP，JFA，Math. Ann.，Trans. AMS等期刊。我们研究了各种时空流形中几类典型非线性波动方程与色散型方程的小初值解的长时间存在性与渐近行为，时空流形包括外区域、nontrapping渐近欧氏空间、Schwarzschild与Kerr黑洞时空等。我们证明了Schwarzschild或小角动量参数的Kerr黑洞时空中的Strauss猜测/John定理；渐近欧氏微扰时空流形中满足零条件的三维多波速拟线性波动方程小初值解的整体存在性；研究了时空流形中的3维Glassey猜测及高维径向Glassey猜测；探讨并得到了具有混合非线性项半线性波动方程在空间维数不大于三的时候具有小初值整体解的完整判据；拓广薛定谔方程第二型广义Strichartz估计，并用于证明三维Zarkharov系统的低正则小能量散射理论等。