HYPOCOERCIVITY AND GLOBAL HYPOELLIPTICITY FOR THE KINETIC FOKKER-PLANCK EQUATION IN Hk SPACES

The purpose of this paper is to extend the hypocoercivity results for the kinetic Fokker-Planck equation in H1 space in Villani’s memoir [45] to higher order Sobolev spaces. As in the L2 and H1 setting, there is lack of coercivity in Hk for the associated operator. To remedy this issue, we shall modify the usual Hk norm with certain well-chosen mixed terms and with suitable coefficients which are constructed by induction on k. In parallel, a similar strategy but with coefficients depending on time (c.f. [34]), usually referred as H´erau’s method, can be employed to prove global hypoellipticity in Hk. The exponents in our regularity estimates are optimal in short time. Moreover, as in our recent work [30], the general results here can be applied in the mean-field setting to get estimates independent of the dimension; in particular, an application to the Curie-Weiss model is presented.