The Gevrey Hypoellipticity for Linear and Non-linear Fokker-planck Equations

Recently, a lot of progress has been made on the study for the spatially homogeneous Boltzmann equation without angular cutoff, cf. [2, 3, 8, 22] and references therein, which shows that the singularity of collision cross-section yields some gain of regularity in the Sobolev space frame on weak solutions for Cauchy problem. That means, this gives the C regularity of weak solution for the spatially homogeneous Boltzmann operator without angular cutoff. The local solutions having the Gevrey regularity have been constructed in [21] for initial data having the same Gevrey regularity, and a genearal Gevrey regularity results have given in [17] for spatially homogeneous and linear Boltzmann equation of Cauchy problem for any initial data. In the other word, there is the smoothness effet similary to heat equation. However, there is no general theory for the spatially inhomogeneous problems. It is now a kinetic equation in which the diffusion part is nonlinear operator of velocity variable. In [1], by using the uncertainty principle and microlocal analysis, they obtain a C regularity results for linear spatially inhomogeneous Boltzmann equation without angular cutoff. In this paper, we will study the Gevrey regularity of weak solution for the the following FokkerPlanck operator in R