Well-posedness of the Fractional Fokker-Planck Equation

TL;DR

This paper proves the well-posedness of the Fractional Fokker-Planck Equation using Schauder estimates, providing insights applicable to broader kinetic models like the Boltzmann and Landau equations.

Contribution

It introduces a Schauder-type estimate approach to establish well-posedness for the FFPE, advancing analytical methods in kinetic theory.

Findings

Established well-posedness of the FFPE

Demonstrated effectiveness of Schauder estimates in kinetic equations

Potential applications to Boltzmann and Landau equations

Abstract

In this paper, we employ a Schauder-type estimate method, as developed in \cite{CHN}, to establish critical well-posedness result for the Fractional Fokker-Planck Equation. This equation serves as a fundamental model in kinetic theory and can be regarded as a semi-linear analogue of the non-cutoff Boltzmann equation. We demonstrate that the techniques introduced in this study are not only effective for the FFPE but also hold promise for broader applications, particularly in addressing the non-cutoff Boltzmann equation and the Landau equation. Our results contribute to a deeper understanding of the analytical framework required for these complex kinetic models.