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

TL;DR

This paper establishes the existence, uniqueness, and regularity of weak solutions for Kolmogorov-Fokker-Planck equations with diverse diffusion types and minimal assumptions, advancing the mathematical understanding of these equations.

Contribution

It introduces new sharp kinetic embeddings and transfer-of-regularity results formulated in a scale-invariant manner for broad classes of Kolmogorov-Fokker-Planck equations.

Findings

Proved existence and uniqueness of weak solutions.

Established regularity results under minimal assumptions.

Developed scale-invariant kinetic embedding tools.

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 \`a la Lions and transfer-of-regularity results \`a la Bouchut--H\''ormander. We formulate these tools in a homogeneous, scale-invariant form, available for a large range of regularity parameters.