133 research outputs found

### Unstable and Stable Galaxy Models

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 $f_{0}(E)$, for which the distribution function $f_{0}$
depends on the particle energy $E$ only. In the first part of the article, we
derive the first sufficient criterion for linear instability of $f_{0}(E):$
$f_{0}(E)$ is linearly unstable if the second-order operator $A_{0}\equiv-\Delta+4\pi\int f_{0}^{\prime}(E)\{I-\mathcal{P}\}dv$ has a
negative direction, where $\mathcal{P}$ is the projection onto the function
space $\{g(E,L)\},$ $L$ 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 $A_{0}$ 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

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 $\kappa>0$.
We prove their linear instability when $\kappa$ 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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