中国科学院数学与系统科学研究院
数学研究所
调和分析及其应用研究中心
学术报告
调和分析和偏微分方程研讨班
Speaker: Professor Emmanuel Trelat （Sorbonne Université）
Title:From microscopic to macroscopic scale equations: mean field, hydrodynamic and graph limits
Time&Venue: 2024年12月13日（星期五）9:30-10:30 & S813
Abstract: Considering finite particle systems, we elaborate on various ways to pass to the limit as the number of agents tends to infinity, either by mean field limit, deriving the Vlasov equation, or by hydrodynamic or graph limit, obtaining the Euler equation. We provide convergence estimates. We also show how to pass from Liouville to Vlasov or to Euler by taking adequate moments. Our results encompass and generalize a number of known results of the literature.As a surprising consequence of our analysis, we show that sufficiently regular solutions of any quasilinear PDE can be approximated by solutions of systems of N particles, to within 1/log(log(N)).This is a work with Thierry Paul.
Speaker: Shengquan Xiang（Peking University）
Title: Exponential mixing for random nonlinear wave equations: weak dissipation and localized control
Time&Venue: 2024年12月13日（星期五）10:30-11:30 & S813
Abstract: We establish a new criterion for exponential mixing and large deviations of random dynamical systems. Our criterion is applicable to a wide range of systems, including in particular dispersive equations. Its verification is in nature related to several topics, i.e., asymptotic compactness in dynamical systems, global stability of evolution equations, and localized control problems. As an initial application, we exploit the exponential mixing and large deviations of random nonlinear wave equations with degenerate damping, critical nonlinearity, and physically localized noise. Based on joint works with Yuxuan Chen, Ziyu Liu, Dongyi Wei, Zhifei Zhang, and Jia-Cheng Zhao.
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