弯曲流形中的Strauss猜测及其相关数学问题

在本项目中, 我们计划研究弯曲流形中的Strauss猜测及其相关的非线性波动方程与色散型方程, 探讨方程的非线性程度、初值的正则性和尺度、背景流形的几何以及解的衰减性与局部和整体存在性之间的深层联系。我们将运用函数空间、调和分析、微局部分析、向量场方法等现代分析技术，挖掘方程的线性与非线性结构，研究渐近平坦流形中各类波动方程的整体适定性问题，如高维外区域的Strauss猜测、阻尼波动方程、具有耦合非线性项的方程组与Glassey猜测。我们将利用算子谱分析技术、椭圆理论，结合广义相对论，研究渐近欧氏空间和黑洞时空中Strauss猜测与相关方程的生命跨度估计。我们还计划研究包括双曲空间和渐近双曲空间在内的弯曲流形中的线性与非线性波动方程：如线性波动方程的色散估计、时空估计；双曲空间中的Glassey猜测、波映照；渐近双曲空间中波动方程、Klein-Gordon方程相应的Strauss猜测。

In this project, we will study the Strauss conjecture and related nonlinear wave equations and dispersive equations, posed on curved manifolds. We will try to study the deep connections between the nonlinear structure, the regularity and size of the initial data, the geometry of background manifolds, the dispersive property and the local/global existence of the solutions for the equations. We will study global well-posedness for wave equations on asymptotically flat manifolds, including the Strauss conjecture on high dimensional exterior domain, damped wave equations, system of nonlinear wave equations with coupled nonlinearities, as well as the Glassey conjecture. The idea is to exploit the linear and nonlinear structure of the problems, and utilize modern analytic techniques, including function spaces, harmonic analysis, microlocal analysis and vector fields methods. We will also investigate the lifespan estimates, both from above and below, for the Strauss conjecture and related equations, on asymptotically Euclidean space and black hold spacetimes. The basic strategy is to combine the characteristics of the wave equations and general relativity with the elliptic theory, as well as the analysis of the spectrum for operators. Furthermore, we plan to study the linear and nonlinear wave equations on (non-asymptotically flat) curved manifolds, including hyperbolic spaces and asymptotically hyperbolic spaces. The following topics will be investigated: dispersive and spacetime estimates for the linear wave equations; the Glassey conjecture and wave maps on hyperbolic spaces; the analog of the Strauss conjecture for the wave and Klein-Gordon equations on asymptotically hyperbolic spaces.