Abstract: We considered the linear stability problem for three-dimensional (3D) states of dynamic equilibrium of a boundless collisionless self-gravitating Vlasov-Poisson gas. Through the replacement of independent variables in the form of hydrodynamic substitution, a transition from the kinetic equations to an infinite system of gas-dynamic equations in the “vortex shallow water” and Boussinesq approximations was made. With the help of direct Lyapunov method, it was shown that the 3D dynamic states of local thermodynamic equilibria of the Vlasov-Poisson gas are absolutely unstable in relation to small 3D perturbations. 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 3D perturbations. Also, the constructive sufficient conditions for linear practical instability of the studied dynamic states of local thermodynamic equilibria in relation to 3D perturbations were obtained, an a priori exponential estimate from below was found, and initial data were described for small 3D perturbations increasing in time. As a means of confirming the results obtained, a series of analytical examples of the considered dynamic equilibrium states were constructed along with small 3D perturbations superimposed on them and growing over time as identified by the found estimate.

Cite: Gubarev Y.G. , Sun S.
STUDY OF INSTABILITY FOR THREE-DIMENSIONAL DYNAMIC EQUILIBRIUM STATES OF SELF-GRAVITATING VLASOV-POISSON GAS
Russian-Chinese Conference “Differential and Difference Equations” 02-06 нояб. 2023