On the Stability of the Euler-Poisson Dark-Fluid Model

We present a stability analysis of a dark-fluid model described as a non-relativistic, rotating, non-viscous, self-gravitating fluid. We assume spherical symmetry and model the matter by a polytropic equation of state. The resulting coupled nonlinear partial differential equation system is solved using a self-similar ansatz known from the Guderley-Landau-Stanyukovich problem. We find that three of the four Lyapunov exponents are negative, while only one is positive. This indicates one unstable direction and three contracting directions in phase space. The single positive exponent is relatively small, suggesting that the self-similar solution is only weakly unstable and remains dynamically robust over the investigated interval.