On the nonlinear instability of nonrotating Stars
We study radial nonlinear instability of compactly supported nonrotating equilibria of the three-dimensional Euler--Poisson system with a physical-vacuum boundary. Let $n^u(\mu)$ denote the radial instability index furnished by the turning-point theory of Lin and Zeng. Under general structural assumptions on the pressure law, suppose that $n^u(\mu)>0$ and that the equilibrium is not a mass extremum, $M'(\mu)\neq0$. Conditional on the existence of a sufficiently regular radial solution on the relevant time interval, we establish two nonlinear escape criteria. First, every perturbation with Hamiltonian strictly below that of the equilibrium exits a fixed neighborhood on a logarithmic time scale controlled by the least unstable linear growth rate. For the subclass of data for which the associated Lyapunov functional is initially nonnegative, we also obtain an explicit exponential lower bound in the weighted displacement norm. Second, sufficiently small perturbations whose Riesz projection onto the finite-dimensional unstable subspace is not too small escape on a logarithmic time scale. The second argument uses the exponential trichotomy of the linearized Hamiltonian flow and an invariant-cone estimate. In the polytropic class, the conditional estimates combine with the radial physical-vacuum local theory to yield unconditional nonlinear instability in the corresponding classical-solution topology. The results complement Jang's nonlinear instability theorem for Lane--Emden stars by treating mechanisms that do not require initial alignment with a leading growing eigenmode and by applying, conditionally, to unstable branches for general equations of state.