Kinetic maximal $L^p_μ(L^p)$-regularity for the fractional Kolmogorov equation with variable density
Lukas Niebel
TL;DR
The article develops a kinetic maximal $L^p_\mu$-regularity theory for nonlocal fractional Kolmogorov equations with density-dependent operators of order $\beta\in(0,2]$, under Hölder-type continuity along characteristics. It identifies the domain $D(A)=H_v^{\beta,p}$ and the relevant trace/kinetic Besov spaces, and proves that, with a suitable regularity of the density along characteristics, the operator family $A(t)$ admits kinetic maximal $L^p_\mu(L^p)$-regularity. This framework enables short-time existence results for strong $L^p_\mu$-solutions to quasilinear fractional kinetic PDEs, via a freezing/coefficient-perturbation strategy and a meticulous partition-of-unity argument. The work extends parabolic and kinetic maximal regularity theory to nonlocal, variable-density operators and provides a robust tool for analyzing nonlinear kinetic equations, including models inspired by Boltzmann-type dynamics.
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 $L^p_μ$-regularity under suitable assumptions on the density and on $p$ and $μ$. We apply this result to prove short-time existence of strong $L^p_μ$-solutions to quasilinear fractional kinetic partial differential equations.