Kinetic maximal $L^p_\mu(L^p)$-regularity for the fractional Kolmogorov equation with variable density

TL;DR

This paper establishes kinetic maximal regularity for a fractional Kolmogorov equation with variable density, enabling the analysis of short-time solutions to related quasilinear fractional kinetic PDEs.

Contribution

It proves kinetic maximal $L^p_^p__$-regularity for a class of fractional Kolmogorov equations with variable density, extending regularity theory to non-local operators with variable coefficients.

Findings

Proves kinetic maximal $L^p_^p__$-regularity under certain conditions.

Establishes short-time existence of strong solutions for quasilinear fractional kinetic PDEs.

Extends regularity results to equations with non-local operators and variable density.

Abstract

We consider the Kolmogorov equation, where the right-hand side is given by a non-local integro-differential operator comparable to the fractional Laplacian in velocity with possibly time, space and velocity dependent density. We prove that this equation admits kinetic maximal $L^p_\mu$-regularity under suitable assumptions on the density and on $p$ and $\mu$. We apply this result to prove short-time existence of strong $L^p_\mu$-solutions to quasilinear fractional kinetic partial differential equations.