Modified scattering dynamics in the Vlasov-Poisson equation near an attractive point mass

TL;DR

This paper investigates the long-term behavior of radially symmetric solutions to the Vlasov-Poisson equation with an attractive point mass, revealing modified scattering dynamics in a low regularity setting.

Contribution

It demonstrates global unique solutions exhibiting modified scattering near an attractive point mass without requiring derivative control, extending understanding of low regularity regimes.

Findings

Solutions asymptotically undergo modified scattering

Global in time, unique Lagrangian solutions are established

Results apply in low regularity settings without derivative control

Abstract

We study the long-time behavior of radially symmetric solutions to the Vlasov-Poisson equation consisting of an attractive point mass and a small, suitably localized and absolutely continuous distribution of particles: if the latter is initially localized on hyperbolic trajectories for the associated Kepler problem, we obtain global in time, unique Lagrangian solutions that asymptotically undergo a modified scattering dynamics (in the sense of distributions). A key feature of this result is its low regularity regime, which does not make use of derivative control, but can be upgraded to strong solutions and strong convergence by propagation of regularity.