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中科院数学与系统科学研究院

数学研究所

华罗庚数学重点实验室

华罗庚青年数学论坛

专题报告

 

 

 

报告人:Prof. Hao Jia(University of Minnesota)

题  目:Soliton resolution for energy critical wave equations

时  间:2018.05.22(星期二),10:00-12:00

地  点:数学院南楼N212室

Abstract:In this talk we will discuss some recent progresses on the study of dynamics of energy critical wave equations, specifically on the soliton resolution conjecture (SRC). SRC predicts that for many dispersive equations, generic solutions should asymptotically de-couple into solitary waves and radiation as time goes to infinity. The conjecture is open for most equations except integrable ones, but is better understood in the case of energy critical wave equations. We will give a sketch of the proof of this conjecture for a sequence of times, in the case of semilinear wave equations. The proof uses many ideas, including optimal perturbation theory, monotonicity formula, unique continuation property for elliptic equations, and most interestingly a channel of energy argument for outgoing waves.

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报告人: Prof. Hao Jia(University of Minnesota)

题  目:Soliton resolution in dispersive equations

时  间:2018.05.24(星期四),10:30-11:30

地  点:数学院南楼N913室

Abstract:A remarkable feature for dispersive equations is the ``simplification" of solutions at large times. For linear dispersive equations, this is well understood. But for nonlinear equations where there are complicated solitary waves, the mechanism by which the solution ``de-couple" into the solitary waves plus radiation is still mysterious, except for integrable systems. We will review some history on this fascinating topic, and explain some recent progress in the energy critical wave equations, such as defocusing wave with potential, focusing wave equations, and wave maps.

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报告人:Prof. Hao Jia(University of Minnesota)

题  目: On the De Gregorio model for 3D Euler equations

时  间:2018.05.25(星期五),10:00-12:00

地  点:数学院南楼N913室

Abstract:The global regularity problem for 3D Euler equations is an important open problem in PDEs. The main issue is to control vorticity, which could grow due to a stretching term in the equation. The main difficulty is to understand the interplay between the vorticity transportation and vorticity stretching. De Gregorio proposed a one dimensional model, based on a modification of the famous Constantin-Lax-Majda model, to gain insight on this effect. It turns out that this one dimensional model is very interesting. Numerical simulations show global existence, but we do not have a proof. In this talk, we will give a proof of global existence in the perturbative regime near the ground state. The proof reveals some interesting features which are relevant in the large data case as well. It also reveals the distinction between several notions of ``criticality" for some quasilinear equations: critical space for well-posedness, persistence of regularity, and the critical space for global existence and long time behavior.

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