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