TY - JOUR

T1 - Study of Instability for One-Dimensional Dynamic Equilibrium States of Self-Gravitating Vlasov–Poisson Gas

AU - Gubarev, Yu G.

AU - Sun, S.

N1 - FUNDING: This work was supported partially by China Scholarship Council (National construction of high-level university public graduate project).

PY - 2022/12

Y1 - 2022/12

N2 - Abstract: 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.

AB - Abstract: 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.

KW - Antonov criterion

KW - Lyapunov functional

KW - Vlasov–Poisson equations

KW - a priori estimate

KW - analytical examples

KW - differential inequality

KW - direct Lyapunov method

KW - gas-dynamic equations

KW - hydrodynamic substitution

KW - instability

KW - small perturbations

KW - stationary solutions

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85150457202&origin=inward&txGid=0250221d07aac6719436d9f91cc1ae71

UR - https://www.elibrary.ru/item.asp?id=50767192

UR - https://www.mendeley.com/catalogue/c08f381c-77b6-398e-9afd-91cbf5b912ed/

U2 - 10.1134/S1995080222150100

DO - 10.1134/S1995080222150100

M3 - Article

VL - 43

SP - 3478

EP - 3485

JO - Lobachevskii Journal of Mathematics

JF - Lobachevskii Journal of Mathematics

SN - 1995-0802

IS - 12

ER -