Damping Versus Oscillations for a Gravitational Vlasov–Poisson System

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 \(k>1/2\), so that the phase space density of the steady state is \(C^1\) at the vacuum boundary if and only if \(k>1\). We prove the following sharp dichotomy result: if \(k>1\), the linear perturbations Landau damp and if \(1/2< k\le 1\) 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 \(k>1\) 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.