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.
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Diacu, F., Ibrahim, S., Lind, C. et al. The Vlasov–Poisson System for Stellar Dynamics in Spaces of Constant Curvature. Commun. Math. Phys. 346, 839–875 (2016). https://doi.org/10.1007/s00220-016-2608-9
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DOI: https://doi.org/10.1007/s00220-016-2608-9