题目一：Mathematical Models and Numerical Simulation for Bose-Einstein Condensation

内容简介：The achievement of Bose-Einstein condensation (BEC) in ultracold vapors of alkali atoms has given enormous impulse to the theoretical and experimental study of dilute atomic gases in condensed quantum states inside magnetic traps and optical lattices.In this talk, I will present a short survey on mathematical models and theories as well as numerical methods for BEC based on the mean field theory.We start with the Gross-Pitaevskii equation (GPE) in three dimensions (3D) for modeling one-component BEC of the weakly interacting bosons,scale it to obtain a three-parameter model and show how to reduce it to two dimensions (2D) and one dimension (1D) GPEs in certain limiting regimes. Mathematical theories and numerical methods for ground states and dynamics of BEC are provided.Extensions to GPE with an angular momentum rotation term for a rotating BEC,to GPE with long-range anisotropic dipole-dipole interaction for a dipolar BEC and to coupled GPEs for spin-orbit coupled BECs are discussed.Finally, some conclusions are drawn and future research perspectives are discussed.

报告人：新加坡国立大学  包伟柱  教授

时  间：2017年12月15日（周五）下午4：00始

题目二：Lipschitz metric for a nonlinear wave equation

内容简介：In this talk, we will discuss a recent breakthrough addressing the Lipschitz continuous dependence of solutions on initial data for a quasi-linear wave equation u_{tt} - c(u)[c(u)u_x]_x = 0. Our earlier results showed that this equation determines a unique flow of conservative solution within the natural energy space H^1(R). However, this flow is not Lipschitz continuous with respect to the H^1 distance, due to the formation of singularity first found by Glassy-Hunter-Zheng. To prove the desired Lipschitz continuous property, we construct a new Finsler type metric, where the norm of tangent vectors is defined in terms of an optimal transportation problem. For paths of piece-wise smooth solutions, we carefully estimate how the distance grows in time. To complete the construction, we prove that the family of piece-wise smooth solutions is dense, following by an application of the Thom's transversality theorem. This is a collaboration work with Alberto Bressan.

报告人：美国堪萨斯大学  陈庚  助理教授

时  间：2017年12月15日（周五）下午5：00始

地  点：南海楼224室

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信息科学技术学院/网络空间安全学院

2017年12月14日