Structure of the Linearized Gravitational Vlasov-Poisson System Close to a Polytropic Ground State

We deal in this paper with a generalized gravitational Vlasov–Poisson system that covers the three- and four-dimensional cases as well as the three-dimensional ultrarelativistic case. This system admits polytropic stationary solutions which are orbitally stable. We study in this paper the linear system obtained after a linearization close to these ground states and prove that the linearized flow displays at most algebraic instabilities. The heart of the proof is the derivation of a positivity property for the linearized Hamiltonian that implies a “quantitative” proof of the orbital stability statement. Our strategy follows the analysis by Weinstein [SIAM J. Math. Anal., 16 (1985), pp. 472–491], who obtained similar results for the nonlinear Schrodinger equation that turned out to be fundamental preliminary properties for the further description of the fine qualitative properties of the Hamiltonian system.

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