Rice University Department of Mathematics Colloquium

The dynamics of Type II solutions to energy critical wave equations

4:00 pm Thursday, November 19, 2015
Hao Jia (U. Chicago)

Abstract: The study of dynamics of energy critical wave equations has seen remarkable progresses in recent years, resulting in deeper understanding of the singularity formation, soliton dynamics, and global large data theory. I will firstly review some of the landmark results, with emphasis on the channel of energy inequalities discovered by Duyckaerts, Kenig and Merle. Applications in the study of global dynamics of defocusing energy critical wave equation with a trapping potential in the radial case will be presented in some detail. We remark that the channel of energy argument provides crucial control on the global dynamics of the solution, and seems to be the only tool currently available to measure dispersion in this context, when we do not assume any smallness condition. The channel of energy argument is however sensitive to dimensions, and in higher dimensions, it is less powerful. We will discuss a new approach to eliminate the dispersive energy when the channel of energy argument fails. Lastly, a new Morawetz estimate in the context of focusing energy critical wave equations will be briefly mentioned. This estimate allows us to study the singularity formation in more details in the nonradial case, without size restriction. As a result, we can characterize the solution along a sequence of times approaching the singular time, up to every nontrivial scale, as modulated solitons. The talk is based on joint works with C. Kenig, and with B.P. Liu, W. Schlag, G.X. Xu.

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