Improving the t^{-2} decay rate for pure transport without a central point mass

Determine whether the algebraic t^{-2} decay-in-time rate established for macroscopic quantities (such as the gravitational force and spatial density) in the radial pure transport dynamics of the three-dimensional gravitational Vlasov–Poisson system without a central Kepler point mass can be improved to a faster decay rate, for suitable regular initial data.

Background

The paper proves quantitative decay rates for the linearised Vlasov–Poisson system around compactly supported equilibria in the presence of a central point mass, obtaining decay that improves with regularity via a limiting absorption principle and resolvent estimates.

In contrast, prior work on the pure transport case (without the point mass) established a limited algebraic decay rate of order t^{-2} for macroscopic quantities in the radial setting. The authors highlight that, in the absence of the central point mass, it is not known whether the t^{-2} rate can be strengthened, underscoring a gap between the pure transport setting and the point-mass-influenced dynamics studied in this paper.