The Vlasov-Poisson System for Stellar Dynamics in Spaces of Constant Curvature

TL;DR

This paper extends the Vlasov-Poisson system to spaces of constant curvature, analyzing well-posedness and stability, including Landau damping, for stellar dynamics on curved geometries like spheres and hyperbolic spaces.

Contribution

It introduces a natural extension of the Vlasov-Poisson system to curved spaces and analyzes the well-posedness and stability of the resulting equations.

Findings

Derived Penrose-type stability conditions.

Established well-posedness of the extended system.

Analyzed Landau damping in curved geometries.

Abstract

We obtain a natural extension of the Vlasov-Poisson system for stellar dynamics to spaces of constant Gaussian curvature $\kappa\ne 0$: the unit sphere $\mathbb S^2$, for $\kappa>0$, and the unit hyperbolic sphere $\mathbb H^2$, for $\kappa<0$. These equations can be easily generalized to higher dimensions. When the particles move on a geodesic, the system reduces to a 1-dimensional problem that is more singular than the classical analogue of the Vlasov-Poisson system. In the analysis of this reduced model, we study the well-posedness of the problem and derive Penrose-type conditions for linear stability around homogeneous solutions in the sense of Landau damping.