Global $L_p$ estimates for kinetic Kolmogorov-Fokker-Planck equations in divergence form

TL;DR

This paper establishes $L_p$ estimates and solvability results for divergence form kinetic Kolmogorov-Fokker-Planck equations in mixed-norm spaces, covering nondegenerate and relativistic cases.

Contribution

It provides the first a priori estimates and solvability results in mixed-norm Lebesgue spaces for divergence form KFP equations with VMO coefficients.

Findings

Established $L_p$ estimates for kinetic KFP equations in divergence form.

Proved unique solvability in mixed-norm Lebesgue spaces.

Extended results to relativistic KFP equations.

Abstract

We present a priori estimates and unique solvability results in the mixed-norm Lebesgue spaces for kinetic Kolmogorov-Fokker-Planck (KFP) equation in divergence form. The leading coefficients are bounded uniformly nondegenerate with respect to the velocity variable $v$ and satisfy a vanishing mean oscillation (VMO) type condition. We consider the $L_2$ case separately and treat more general equations which include the relativistic KFP equation.