Global Existence And Decay Rates Of The Solutions Near Maxwellian For Non-linear Fokker--Planck Type Equations

The purpose of this paper is to study the nonlinear Fokker-Planck equation and the nonlinear Vlasov-Poisson-Fokker-Planck solution.The research includes the following two aspects:First,we study the global existence and decay rates of the solutions near Maxwellian for non-linear Fokker-Planck equations in the whole space.The global existence is proved by combining uniform-in-time energy estimates and local solution constructed by Picard type iteration.The decay rates of the nonlinear model is obtained by using the precise spectral analysis of the linearized Fokker-Planck operator as well as the energy method.The nonlinearity in the model brings new difficulty for the energy estimates,which is resolved by additional tailored weighted-in-v energy estimates suitable for Fokker-Planck operators.Second,we establish the exponential time decay of smooth solutions around a global Maxwellian to the non-linear Vlasov-Poisson-Fokker-Planck equations in the whole space by uniform-in-time energy estimates.The non-linear coupling of macroscopic part and Fokker--Planck operator in the model brings new difficulties for the energy estimates,which can also be solved by the above method.