On quantitative linear gravitational relaxation

TL;DR

This paper establishes quantitative decay rates for the linearised Vlasov-Poisson system around compact equilibria, demonstrating a gravitational analogue of linear Landau damping in a radial setting.

Contribution

It introduces new analytical techniques to prove linear asymptotic stability and decay in a gravitational context, overcoming obstacles posed by stable trapping.

Findings

Proves decay of gravitational potential in the linearized Vlasov-Poisson system.

First linear asymptotic stability result for such gravitational equilibria.

Develops new methods including normal form and resolvent bounds.

Abstract

We prove quantitative decay rates for the linearised Vlasov-Poisson system around compactly supported equilibria. More precisely, we prove decay of the gravitational potential induced by the radial dynamics of this system in the presence of a point mass source. Our result can be interpreted as the gravitational version of linear Landau damping in the radial setting and hence the first linear asymptotic stability result around such equilibria. We face fundamental obstacles to decay caused by the presence of stable trapping in the problem. To overcome these issues we introduce several new ideas. We use different tools, including the Birkhoff-Poincar\'e normal form, action-angle type variables, and delicate resolvent bounds to prove a suitable version of the limiting absorption principle and obtain the decay-in-time.