Topic:The evolution to equilibrium of solutions to nonlinear Fokker-Planck equations
Speaker: Professor MichaelRöckner
Event date: 5/27/2019
Event time: 14:30 p.m.
Venue: Lecture Hall1506, Building 9
Sponsor: School of Mathematics and Statistics, Institute of Science and Technology
Abstract: The talk isabout the so-called H -Theorem for a class of nonlinear Fokker-Planck equationswhich are of porous media type on the whole Euclidean space perturbed by atransport term. We first construct a solution in the sense of mild solutions onL^1 through a nonlinear semigroup of contractions. Then we study the asymptoticbehavior of the solutions when time tends to infinity. For a large class M ofinitial conditions we show their relative compactness with respect to local L^1convergence, while all limit points belong to L^1. Under an additionalassumption we obtain that we in fact have convergence in L^1, if the initial condition is a probability density.The limit is then identified as the unique stationary solution in M to thenonlinear Fokker-Planck equation. This solution is thus an invariant measure ofthe solution to the corresponding distribution dependent SDE whose timemarginals converge to it in L^1. It turns out that under our conditions theunderlying nonlinear Kolmogorov operator is a (both in the second and firstorder part) nonlinear analog of the generator of a distorted Brownian motion.The solution of the above mentioned distribution dependent SDE can thus beinterpreted as a “nonlinear distorted Brownian motion“. Our main technique forthe proofs is to construct a suitable Lyapunov function acting nonlinearly onthe path in L^1, which is given by the nonlinear contraction semigroup applied to the initial condition, and thenadapt a classical technique of Pazy to our situation. This Lyapunov function isgiven by a generalized entropy function (which in the linear case specializesto the usual Boltzmann-Gibbs entropy) plus a mean energy part.