Existence and uniqueness of solutions for the Vlasov-Fokker-Planck equation in the two-dimensional space

We consider the Cauchy problem for the Fokker-Planck equation associated with the Vlasov-Poisson system for interacting particles in a thermal reservoir. We describe the physical model from which it can be heuristically derived, both for Newtonian forces in the gravitational case and for Coulomb forces in the case of electric fields. We present the state-of-the-art existence theory depending on the regularity of the initial data and the singularity of the potential term, since the difficulty increases with the dimension of the physical space for the underlying nonlinear diffusion process. Then, we give a detailed account of the existence results in the two-dimensional space according to [H. Neunzert, M. Pulvirenti, L. Triolo, On the Vlasov-Fokker-Planck equation, Math. Meth. Appl. Sci. 6 (1984), 527-538] Finally, we discuss a counter-example to the global existence for the stellar dynamics in the four-dimensional space