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

TL;DR

This paper establishes global $L_p$ estimates for degenerate kinetic Kolmogorov-Fokker-Planck equations in nondivergence form with VMO coefficients, providing solvability results and applications to the Landau equation without kernel estimates.

Contribution

It introduces new global $L_p$ estimates for kinetic equations with VMO coefficients in nondivergence form, extending previous results to more general coefficient regularity.

Findings

Proved global a priori estimates in weighted mixed-norm Lebesgue spaces.

Established solvability results for the kinetic equations.

Applied the main results to the Landau equation.

Abstract

We study the degenerate Kolmogorov equations (also known as kinetic Fokker-Planck equations) in nondivergence form. The leading coefficients $a^{ij}$ are merely measurable in $t$ and satisfy the vanishing mean oscillation (VMO) condition in $x, v$ with respect to some quasi-metric. We also assume boundedness and uniform nondegeneracy of $a^{ij}$ with respect to $v$. We prove global a priori estimates in weighted mixed-norm Lebesgue spaces and solvability results. We also show an application of the main result to the Landau equation. Our proof does not rely on any kernel estimates.