133 research outputs found

    Unstable and Stable Galaxy Models

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    To determine the stability and instability of a given steady galaxy configuration is one of the fundamental problems in the Vlasov theory for galaxy dynamics. In this article, we study the stability of isotropic spherical symmetric galaxy models f0(E)f_{0}(E), for which the distribution function f0f_{0} depends on the particle energy EE only. In the first part of the article, we derive the first sufficient criterion for linear instability of f0(E):f_{0}(E): f0(E)f_{0}(E) is linearly unstable if the second-order operator A0β‰‘βˆ’Ξ”+4Ο€βˆ«f0β€²(E){Iβˆ’P}dv A_{0}\equiv-\Delta+4\pi\int f_{0}^{\prime}(E)\{I-\mathcal{P}\}dv has a negative direction, where P\mathcal{P} is the projection onto the function space {g(E,L)},\{g(E,L)\}, LL being the angular momentum [see the explicit formula (\ref{A0-radial})]. In the second part of the article, we prove that for the important King model, the corresponding A0A_{0} is positive definite. Such a positivity leads to the nonlinear stability of the King model under all spherically symmetric perturbations.Comment: to appear in Comm. Math. Phy

    Stability and instability of self-gravitating relativistic matter distributions

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    We consider steady state solutions of the massive, asymptotically flat, spherically symmetric Einstein-Vlasov system, i.e., relativistic models of galaxies or globular clusters, and steady state solutions of the Einstein-Euler system, i.e., relativistic models of stars. Such steady states are embedded into one-parameter families parameterized by their central redshift ΞΊ>0\kappa>0. We prove their linear instability when ΞΊ\kappa is sufficiently large, i.e., when they are strongly relativistic, and that the instability is driven by a growing mode. Our work confirms the scenario of dynamic instability proposed in the 1960s by Zel'dovich \& Podurets (for the Einstein-Vlasov system) and by Harrison, Thorne, Wakano, \& Wheeler (for the Einstein-Euler system). Our results are in sharp contrast to the corresponding non-relativistic, Newtonian setting. We carry out a careful analysis of the linearized dynamics around the above steady states and prove an exponential trichotomy result and the corresponding index theorems for the stable/unstable invariant spaces. Finally, in the case of the Einstein-Euler system we prove a rigorous version of the turning point principle which relates the stability of steady states along the one-parameter family to the winding points of the so-called mass-radius curve.Comment: 92 pages; several proofs are revised and some previous errors correcte
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