Hidden symmetries and decay for the Vlasov equation on the Kerr spacetime

TL;DR

This paper establishes energy decay and boundedness for solutions of the massless Vlasov equation in slowly rotating Kerr spacetime, extending techniques from wave equations and vector-field methods to relativistic kinetic theory.

Contribution

It introduces a novel combination of methods to prove decay and boundedness of the Vlasov equation on Kerr spacetime, a significant step in understanding relativistic kinetic equations in curved backgrounds.

Findings

Proves bounded energy for Vlasov solutions in Kerr spacetime

Establishes integrated energy decay for the Vlasov equation

Extends vector-field method to relativistic kinetic equations

Abstract

This paper proves the existence of a bounded energy and integrated energy decay for solutions of the massless Vlasov equation in the exterior of a very slowly rotating Kerr spacetime. This combines methods previously developed to prove similar results for the wave equation on the exterior of a very slowly rotating Kerr spacetime with recent work applying the vector-field method to the relativistic Vlasov equation.