Weak solutions to Kolmogorov-Fokker-Planck equations: regularity, existence and uniqueness

Abstract

We prove existence, uniqueness and regularity of weak solutions of Kolmogorov--Fokker--Planck equations with either local or non-local diffusion in the velocity variable and rough diffusion coefficients or kernels. Our results cover the Cauchy problem and allow a broad class of source terms under minimal assumptions. The core of the analysis is a set of sharp kinetic embeddings à la Lions and transfer-of-regularity results à la Bouchut--H\''ormander. We formulate these tools in a homogeneous, scale-invariant form, available for a large range of regularity parameters.

arXiv commentThis version contains the same main results but has been reorganized in a new presentation of them, making their validity for all parameters beta>0 more explicit. Some proofs were adjusted to that change. Some corollaries are explicitly stated in the introduction and more explanation are given. A density lemma (section 2) is added and an inaccuracy in Section 8 eliminated. The modifications here have no impact on the follow-up on the construction of fundamental solutions that is already published in SIAM J. of Mathematical Analysis