中科院数学与系统科学研究院

数学研究所

偏微分方程研讨会

 

报告人:刘保平(北京大学)

  目:Global center stable manifold for nonlinear wave equation with potential

  间:2021.09.24(星期五),15:00-16:00

  点:数学院南楼N913

  要:In this talk, we consider the defocusing energy critical wave equation with a trapping potential. When the potential decays fast enough, it is easy to show that all finite energy solutions exist globally, hence our main interest is to describe the long time dynamics. In the radial case, our previous works gave a complete answer and we were able to classify all the long time dynamics. Here we partly extend previous result to the nonradial case, and show that the set of initial data for which solutions scatter to an unstable excited state forms a finite co-dimensional path connected C1 manifold in the energy space. This gives us a better understanding of the non-generic behavior of solutions, with the generic behavior left as an open problem.
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报告人:郑继强(北京应用物理与计算数学研究所)

  目:Dispersive and Strichartz estimate for dispersive equations with scaling -critical electromagnetic potential

  间:2021.09.24(星期五),16:00-17:00

  点:数学院南楼N913

  要:In this talk, We study the dispersive equation with Aharonov-Bohm magnetic potential. We prove sharp time-decay estimates in the purely magnetic case, and Strichartz estimates for the complete model, involving a critical electromagnetic field. The novel ingredients are the Schwartz kernels of the spectral measure and heat propagator of the Schrödinger operator in Aharonov-Bohm magnetic fields. In particular, we explicitly construct the representation of the spectral measure and resolvent of the Schrödinger operator with Aharonov-Bohm potentials, and show that the heat kernel in critical electromagnetic fields satisfies Gaussian boundedness. This talk is based on a series of joint works with Xiaofen Gao, Luca Fanelli, Zhiqing Yin and Junyong Zhang.

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