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.