Study of Instability for One-Dimensional Dynamic Equilibrium States of Self-Gravitating Vlasov–Poisson Gas

In this paper, the linear stability problem for one-dimensional (1D) states of dynamic equilibrium of a boundless collisionless self-gravitating Vlasov–Poisson gas was considered. Using the replacing of independent variables in the form of hydrodynamic substitution, a transition was made from the kinetic equations to an infinite system of gas-dynamic equations in the ‘‘vortex shallow water’’ and Boussinesq approximations. The absolute linear instability for dynamic states of local thermodynamic equilibria of the Vlasov–Poisson gas with respect to 1D perturbations was proved by the direct Lyapunov method. In the process of proving instability, a formal nature of the well-known Antonov criterion for linear stability of dynamic equilibrium states of self-gravitating stellar systems was discovered, so that this criterion is valid only with respect to some incomplete unclosed subclass of small 1D perturbations. Also, the constructive sufficient conditions for linear practical instability of the studied dynamic states of local thermodynamic equilibria with respect to 1D perturbations are obtained, an a priori exponential estimate from below is found, and initial data are described for small 1D perturbations increasing in time. To confirm the results obtained, analytical examples of the studied dynamic equilibrium states and small 1D perturbations superimposed on them, which grow in time according to the found estimate, are constructed.