TY - CHAP

T1 - Study of Linear Stability for Cylindrically Symmetrical States of Dynamic Equilibrium of Two-Component Vlasov–Poisson Plasma

AU - Gubarev, Yuriy G.

AU - Luo, Jingyue

N1 - Conference code: 12

PY - 2024

Y1 - 2024

N2 - We consider the linear stability problem for dynamic equilibria of two-component Vlasov–Poisson plasma in cylindrically symmetrical statement. The hydrodynamic substitution of independent variables is performed in order to transform the Vlasov–Poisson equations to an infinite system of gas-dynamic equations. It is important that exact stationary solutions to gas-dynamic equations are equivalent to exact stationary solutions to the Vlasov–Poisson equations. The sufficient condition of linear stability for exact stationary solutions to the Vlasov–Poisson equations is studied. Previously, this condition was not reversed either for small or, especially, for finite perturbations. To fulfill such reversion in the linear approximation, these gas-dynamic equations are linearized near their exact stationary solutions. The a priori exponential estimate from below is constructed for a subclass of small cylindrically symmetrical perturbations of exact stationary solutions to gas-dynamic equations, which grow over time and are described by the field of Lagrangian displacements. The countable set of sufficient conditions for linear practical instability is obtained. Thus, the Newcomb-Gardner-Rosenbluth sufficient condition for linear stability of exact stationary solutions to the Vlasov–Poisson equations is reversed. Moreover, a formal nature of this condition is revealed with respect to the considered small perturbations. As a result, by the direct Lyapunov method, an absolute instability for exact stationary solutions to the mathematical model of two-component Vlasov–Poisson plasma in relation to small cylindrically symmetrical perturbations is proved.

AB - We consider the linear stability problem for dynamic equilibria of two-component Vlasov–Poisson plasma in cylindrically symmetrical statement. The hydrodynamic substitution of independent variables is performed in order to transform the Vlasov–Poisson equations to an infinite system of gas-dynamic equations. It is important that exact stationary solutions to gas-dynamic equations are equivalent to exact stationary solutions to the Vlasov–Poisson equations. The sufficient condition of linear stability for exact stationary solutions to the Vlasov–Poisson equations is studied. Previously, this condition was not reversed either for small or, especially, for finite perturbations. To fulfill such reversion in the linear approximation, these gas-dynamic equations are linearized near their exact stationary solutions. The a priori exponential estimate from below is constructed for a subclass of small cylindrically symmetrical perturbations of exact stationary solutions to gas-dynamic equations, which grow over time and are described by the field of Lagrangian displacements. The countable set of sufficient conditions for linear practical instability is obtained. Thus, the Newcomb-Gardner-Rosenbluth sufficient condition for linear stability of exact stationary solutions to the Vlasov–Poisson equations is reversed. Moreover, a formal nature of this condition is revealed with respect to the considered small perturbations. As a result, by the direct Lyapunov method, an absolute instability for exact stationary solutions to the mathematical model of two-component Vlasov–Poisson plasma in relation to small cylindrically symmetrical perturbations is proved.

KW - Absolute linear instability

KW - Cylindrically symmetrical dynamic equilibria

KW - Vlasov–Poisson plasma

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85195273602&origin=inward&txGid=de416aa94bffde290b81cb6f271d7544

UR - https://www.mendeley.com/catalogue/7aca4d2c-caaa-389e-b59f-cb08016c9d37/

U2 - 10.1007/978-3-031-52965-8_37

DO - 10.1007/978-3-031-52965-8_37

M3 - Chapter

SN - 9783031529641

T3 - Springer Proceedings in Mathematics and Statistics

SP - 471

EP - 480

BT - Springer Proceedings in Mathematics and Statistics

PB - Springer-Verlag GmbH and Co. KG

T2 - 12th International Conference on Mathematical Modeling in Physical Sciences

Y2 - 28 August 2023 through 31 August 2023

ER -