Speaker: Michael Roeckner

Title: Nonlinear Fokker-Planck-Kolmogorov Equations and Nonlinear Markov Processes: The 2D Vorticity Navier-Stokes Equation

Abstract: Since the middle of last century a substantial part of stochastic analysis has been devoted to the relationship between (parabolic) linear partial differential equations (PDEs), more precisely, linear Fokker-Planck-Kolmogorov equations (FPKEs), and stochastic differential equations (SDEs), or more generally Markov processes. Its most prominent example is the classical heat equation on one side and the Markov process given by Brownian motion on the other. This talk is about the nonlinear analogue, i.e., the relationship between nonlinear FPKEs on the analytic side and McKean-Vlasov SDEs (of Nemytskii-type), or more generally, nonlinear Markov processes in the sense of McKean on the probabilistic side. This program has been initiated by McKean already in his seminal PNASpaper from 1966 and this talk is about recent developments in this field. Topics will include existence and uniqueness results for distributional solutions of the nonlinear FPKEs on the analytic side and equivalently existence and uniqueness results for weak solutions of the McKean-Vlasov SDEs on the probabilistic side. Furthermore, criteria for the corresponding path laws to form a nonlinear Markov process will be presented. Among the applications are e.g. porous media equations (including such with nonlocal operators replacing the Laplacian and possibly being perturbed by a transport term) and their associated nonlinear Markov processes. A special emphasis will lie on the 2D Navier-Stokes equation in vorticity form and its associated nonlinear Markov process.

Joint work with: Viorel Barbu, Al.I. Cuza University and Octav Mayer Institute of Mathematics of Romanian Academy, Ia¸si, Romania Deng Zhang, Shanghai Jiao Tong University.

References on which the talk is based: 

[1] Barbu, V., R¨ockner, M., Nonlinear Fokker–Planck flows and their probabilistic counterparts, Lecture Notes in Mathematics, Springer 2024+, pp. ix + 214 

[2] Barbu, V., R¨ockner, M., Nonlinear Fokker–Planck equations with fractional Laplacian and McKean–Vlasov SDEs with L´evy noise, Probab. Theory Rel. Fields (2024), online