Long Time Dynamics Of Nonlinear Dispersive PDEs And Geometric Flows

In this paper,we study the dynamic behaviors of dispersive partial differential e-quations and dispersive geometric flows.We mainly focus on the soliton resolution for damped Klein-Gordon and long time behaviors of Landau-Lifshitz flows and asymp-totic stability of harmonic maps under the wave map equation.In 2015,I and cowork-ers solved the soliton resolution of damped Klein-Gordon equations in the non-radial case,which generalized the radial result of Burq,Schalg,Raugel to some extend.D-ifferent from the invariant manifolds method used in the radial case,we combine the compact-concentrate attractor technique of Tao with the damping effect and show any global bounded solutions split into the superposition of divergent equilibriums along some subsequence of any given time sequence.In 2016,1 and coworkers obtained the asymptotic stability of small energy harmonic maps under the wave map equation from two dimensional hyperbolic space to two dimensional hyperbolic space.Besides,we studied the long time behaviors of Landau-Lifshitz flow earlier.In these two papers,we construct the caloric gauge when nontrivial harmonic maps occur,and successfully apply it to the asymptotic behavior study.In the introduction,we recall the background material including the physical motivation,the research history and the preparations.In the first chapter,we illustrate the outline of the proof of soliton resolution of damped Klein-Gordon in the non-radial case.In the second chapter,we give a detailed proof of the asymptotic stability of small harmonic maps between 2D hyperbolic spaces under the wave map equation.