Abstract
We consider two classes of steady states of the three-dimensional, gravitational Vlasov-Poisson system: the spherically symmetric Antonov-stable steady states (including the polytropes and the King model) and their plane symmetric analogues. We completely describe the essential spectrum of the self-adjoint operator governing the linearized dynamics in the neighborhood of these steady states. We also show that for the steady states under consideration, there exists a gap in the spectrum. We then use a version of the Birman-Schwinger principle first used by Mathur to derive a general criterion for the existence of an eigenvalue inside the first gap of the essential spectrum, which corresponds to linear oscillations about the steady state. It follows in particular that no linear Landau damping can occur in the neighborhood of steady states satisfying our criterion. Verification of this criterion requires a good understanding of the so-called period function associated with each steady state. In the plane symmetric case we verify the criterion rigorously, while in the spherically symmetric case we do so under a natural monotonicity assumption for the associated period function. Our results explain the pulsating behavior triggered by perturbing such steady states, which has been observed numerically.
Highlights
In a numerical investigation of the EinsteinVlasov system, which is the relativistic version of the Vlasov-Poisson system, it was observed that such perturbations lead to solutions which oscillate about the steady state [1]
While we do not notationally distinguish between a general, or spherically symmetric, or plane symmetric phase space density f, we will notationally distinguish between function spaces and operators consisting of or acting on spherically symmetric, or plane symmetric functions respectively; the latter will be distinguished by a bar, so A will act on spherically symmetric functions, while Awill act on plane symmetric ones
In Theorem 8.13 we prove that there exist classes of planar steady states such that the associated linearized operator has an eigenvalue in the principal gap
Summary
The three-dimensional gravitational Vlasov-Poisson system is the fundamental system of equations used in astrophysics to describe galaxies [8]. In a numerical investigation of the EinsteinVlasov system, which is the relativistic version of the Vlasov-Poisson system, it was observed that such perturbations lead to solutions which oscillate about the steady state [1]. In these oscillations the spatial support of the solutions expands and contracts in a time-periodic way, i.e., after perturbation the state starts to pulse. The same behavior was observed numerically for the Vlasov-Poisson system in [56], and again for the Einstein-Vlasov system in [21] Such pulsating solutions are classical for the Euler-Poisson system and have been used to explain the Cepheid variable stars [15,63]; a mathematically rigorous analysis of these solutions is provided in [31,50]. Of this introduction we outline our paper in more detail and put it into perspective
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