本项目旨在研究双曲几何流的全局动力学, 尤其是弯曲时空上的波映射方程和薛定谔流. 波映射方程是高能物理的基本模型, 薛定谔流方程是磁固体物理最重要的方程, 两者都是典型的双曲几何流. 本项目主要运用调和分析和几何分析的技术来研究渐进平坦和双曲空间上的波映射、 薛定谔流方程的平衡态的稳定性和低正则适定性问题. 调和映射是波映射和薛定谔流的稳态解, 根据气泡定理或者孤子分解猜想, 任意初始值产生的动力学在爆破时刻或者无穷远时刻都归结为一个或多个调和映射附近的动力学, 因此研究调和映射在波映射和薛定谔流下的稳定性具有关键的意义. 特别地, 我们着重探讨底空间的曲率如何影响双曲几何流的长时间行为.

This project aims to study the global dynamics of hyperbolic geometric flows, especially the wave map equations and Schrodinger map flow in curved spacetimes. The wave map equation is the basic model of high-energy physics, and the Schrodinger map flow is the most important equation for magnetic solids physics. Both of them are typical hyperbolic geometric flows. This project mainly uses harmonic analysis and geometric analysis techniques to study the stability of the equilibrium states and low regularity well-posedness for wave maps and Schrodinger map flows on asymptotically flat spaces and hyperbolic spaces. Harmonic maps are the stationary solutions of wave maps and Schrodinger map flows. According to bubbling theorem or soliton resolution conjecture, the dynamics generated by any initial data are finally reduced to the dynamics near one or more harmonic maps at the time of explosion or infinity. Therefore, it is of vital significance to study the stability of harmonic maps under wave maps and Schrodinger map flows. In particular, we mainly explore how the curvatures of the base space affect the long time behaviors of hyperbolic geometric flows.