演講摘要：
We present a unifying theme of uniform-in-time stability from hyperbolic balance laws to kinetic equations and collective dynamics using the nonlinear functional approach. Nonlinear functional approach was first introduced by James Glimm's 1965 work on the Cauchy problem of hyperbolic conservation laws and it was further improved and generalized by Tai-Ping Liu, Tong Yang, Alberto Bressan, Stephano Bianchini and others on the uniform-in-time L¹-stability of entropy weak solutions generated by the Glimm scheme and front-tracking scheme and vanishing viscosity limit problem from viscous conservation laws to hyperbolic conservation laws. In this talk, we will discuss how this nonlinear functional approach can be used in the uniform stability of kinetic equations such as the Boltzmann equation and the Vlasov type equations near vacuum and particle models for collective dynamics. We also provide several applications of nonlinear functionals approach in the uniform-in-time mean field limit and continuous transition from discrete models to continuous models for collective dynamics.

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