报告人 赵立丰（中国科学与技术大学）

 
题目：Long time behavior of nonlinear dispersive equations - from the threshold conjecture to the soliton resolution conjecture（I）

报告时间：2015.11.27 10:00-10:50

 
报告地点：精正楼二楼学术报告厅
 
摘要：The nonlinear dispersive equations are among the fundamental equations in mathematical physics including nonlinear wave equations, Schrodinger equations and KdV equations, etc. They come from various physical background such as quantum field theory, fluid mechanics, optics and general relativity. There has been enormous progress in the mathematical study of these kinds of equations in the last three decades. In this talk, I will focus on the main conjectures in this field and review some important progress. 
In the first part,  I will talk about the threshold conjecture. Roughly speaking, the conjecture states that there is a threshold ,which usually is provided by the ground states, under which all solutions are global and scatters both forward and backward in time. The verification of the conjecture is especially hard for solutions in critical spaces. After explaining the strategy of the induction on energy and compactness-contradiction , I will give some surveys on the progress on this conjecture for various dispersive equations. Some open problems will be mentioned. 
In fact, the verification of the threshold conjecture is just the first step toward the grand conjecture - soliton resolution conjecture. The conjecture predicts the behavior of generic solutions to the nonlinear dispersive equations, especially solutions above the threshold. In the second part,  after stating the conjecture I will talk about the progress on the verification of the conjecture. Some strategies will be explained including the invariant manifold theory and the method of concentration-compact attractors. 

 


报告人 赵立丰（中国科学与技术大学）

 
题目：Long time behavior of nonlinear dispersive equations - from the threshold conjecture to the soliton resolution conjecture（II）

报告时间：2015.11.27 11:00-11:50

 
报告地点：精正楼二楼学术报告厅

 
摘要：The nonlinear dispersive equations are among the fundamental equations in mathematical physics including nonlinear wave equations, Schrodinger equations and KdV equations, etc. They come from various physical background such as quantum field theory, fluid mechanics, optics and general relativity. There has been enormous progress in the mathematical study of these kinds of equations in the last three decades. In this talk, I will focus on the main conjectures in this field and review some important progress. 
In the first part,  I will talk about the threshold conjecture. Roughly speaking, the conjecture states that there is a threshold ,which usually is provided by the ground states, under which all solutions are global and scatters both forward and backward in time. The verification of the conjecture is especially hard for solutions in critical spaces. After explaining the strategy of the induction on energy and compactness-contradiction , I will give some surveys on the progress on this conjecture for various dispersive equations. Some open problems will be mentioned. 
In fact, the verification of the threshold conjecture is just the first step toward the grand conjecture - soliton resolution conjecture. The conjecture predicts the behavior of generic solutions to the nonlinear dispersive equations, especially solutions above the threshold. In the second part,  after stating the conjecture I will talk about the progress on the verification of the conjecture. Some strategies will be explained including the invariant manifold theory and the method of concentration-compact attractors.