On algebraic damping close to inhomogeneous Vlasov equilibria in multi-dimensional spaces

We investigate the asymptotic damping of a perturbation around inhomogeneous stable stationary states of the Vlasov equation in spatially multi-dimensional systems. We show that branch singularities of the Fourier-Laplace transform of the perturbation yield algebraic dampings. In two spatial dimensions, we classify the singularities and compute the associated damping rate and frequency. This 2D setting also applies to spherically symmetric self-gravitating systems. We validate the theory using a toy model and an advection equation associated with the isochrone model, a model of spherical self-gravitating systems.