# Damping Versus Oscillations for a Gravitational Vlasov–Poisson System

## TL;DR

This paper studies the behavior of gravitational systems and finds that their long-term dynamics depend on the smoothness of their initial state.

## Contribution

The paper proves a sharp dichotomy between damping and oscillations based on the regularity of the steady state at the vacuum boundary.

## Key findings

- For k > 1, linear perturbations exhibit Landau damping.
- For 1/2 < k ≤ 1, perturbations do not damp and instead oscillate.
- The absence of embedded eigenvalues in the essential spectrum is a key result.

## Abstract

We consider a family of isolated inhomogeneous steady states of the gravitational Vlasov–Poisson system with a point mass at the centre. These are parametrised by the polytropic index \documentclass[12pt]{minimal}
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\begin{document}$$k>1/2$$\end{document}k>1/2, so that the phase space density of the steady state is \documentclass[12pt]{minimal}
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\begin{document}$$C^1$$\end{document}C1 at the vacuum boundary if and only if \documentclass[12pt]{minimal}
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\begin{document}$$k>1$$\end{document}k>1. We prove the following sharp dichotomy result: if \documentclass[12pt]{minimal}
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\begin{document}$$k>1$$\end{document}k>1, the linear perturbations Landau damp and if \documentclass[12pt]{minimal}
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\begin{document}$$1/2< k\le 1$$\end{document}1/2<k≤1 they do not. The above dichotomy is a new phenomenon and highlights the importance of steady state regularity at the vacuum boundary in the discussion of the long-time behaviour of the perturbations. Our proof of (nonquantitative) gravitational relaxation around steady states with \documentclass[12pt]{minimal}
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\begin{document}$$k>1$$\end{document}k>1 is the first such result for the gravitational Vlasov–Poisson system. The key novelty of this work is the proof that no embedded eigenvalues exist in the essential spectrum of the linearised system.