Kinetic maximal $L^2$-regularity for the (fractional) Kolmogorov equation

TL;DR

This paper introduces kinetic maximal $L^2$-regularity with weights for the fractional Kolmogorov equation, characterizing solution regularity via fractional anisotropic Sobolev spaces and analyzing solution semi-flows and regularization.

Contribution

It defines a new regularity concept for the fractional Kolmogorov equation and characterizes solution spaces, advancing understanding of solution regularity and flow properties.

Findings

Solutions form a semi-flow in a suitable function space

Instantaneous regularization property is established

Characterization of regularity via fractional anisotropic Sobolev spaces

Abstract

We introduce the notion of kinetic maximal $L^2$-regularity with temporal weights for the (fractional) Kolmogorov equation. In particular, we determine the function spaces for the inhomogeneity and the initial value which characterize the regularity of solutions to the fractional Kolmogorov equation in terms of fractional anisotropic Sobolev spaces. It is shown that solutions of the homogeneous (fractional) Kolmogorov equation define a semi-flow in a suitable function space and the property of instantaneous regularization is investigated.