TY - GEN

T1 - On the instability for one subclass of three-dimensional dynamic equilibrium states of the electron Vlasov–Poisson gas

AU - Gubarev, Yuriy

AU - Liu, Yang

N1 - FWGG-2021-0008, FWGG-2021-0004

PY - 2025

Y1 - 2025

N2 - This paper considers the spatial movement of a boundless collisionless electron Vlasov–Poisson gas in the three-dimensional (3D) Cartesian coordinate system. By replacing the independent variables as a hydrodynamic substitution we transform the kinetic equations into the infinite gas-dynamic equations in the “vortex shallow water” and Boussinesq approximations. In the proof of linear instability for exact stationary solutions to the Vlasov–Poisson equations, we reverse the well-known Newcomb–Gardner–Rosenbluth sufficient condition for stability regarding a specific class of small spatial perturbations that are incomplete and unclosed. Additionally, we derive an original linear second-order differential inequality with constant coefficients for the Lyapunov functional. When the conditions established in this paper for linear practical instability of exact stationary solutions are satisfied, we obtain an a priori exponential estimate from below for the growth rate of small 3D perturbations using this inequality. Importantly, since this estimate is derived without imposing additional restrictions on exact stationary solutions, we establish absolute linear instability for spatial dynamic equilibrium states of the electron Vlasov–Poisson gas with respect to 3D perturbations. To confirm the results obtained, analytical examples of the studied equilibrium states and small spatial perturbations superimposed on them, which grow in time according to the found estimate, are constructed for kinetic and gas-dynamic systems.

AB - This paper considers the spatial movement of a boundless collisionless electron Vlasov–Poisson gas in the three-dimensional (3D) Cartesian coordinate system. By replacing the independent variables as a hydrodynamic substitution we transform the kinetic equations into the infinite gas-dynamic equations in the “vortex shallow water” and Boussinesq approximations. In the proof of linear instability for exact stationary solutions to the Vlasov–Poisson equations, we reverse the well-known Newcomb–Gardner–Rosenbluth sufficient condition for stability regarding a specific class of small spatial perturbations that are incomplete and unclosed. Additionally, we derive an original linear second-order differential inequality with constant coefficients for the Lyapunov functional. When the conditions established in this paper for linear practical instability of exact stationary solutions are satisfied, we obtain an a priori exponential estimate from below for the growth rate of small 3D perturbations using this inequality. Importantly, since this estimate is derived without imposing additional restrictions on exact stationary solutions, we establish absolute linear instability for spatial dynamic equilibrium states of the electron Vlasov–Poisson gas with respect to 3D perturbations. To confirm the results obtained, analytical examples of the studied equilibrium states and small spatial perturbations superimposed on them, which grow in time according to the found estimate, are constructed for kinetic and gas-dynamic systems.

KW - Direct Lyapunov method

KW - Instability

KW - Small perturbations

KW - Stationary solutions

KW - Vlasov–Poisson equations

UR - https://www.scopus.com/pages/publications/105021932331

UR - https://www.elibrary.ru/item.asp?id=80667093

UR - https://www.mendeley.com/catalogue/dedddfb6-de2f-36c6-8d26-000c7b231044/

U2 - 10.1515/9783111570518-012

DO - 10.1515/9783111570518-012

M3 - Conference contribution

SN - 9783111570518

SP - 111

EP - 120

BT - Analytical Methods in Differential Equations. De Gruyter Proceedings in Mathematics

PB - Walter de Gruyter GmbH

CY - Berlin

T2 - Всероссийская конференция «Математические проблемы механики сплошных сред», посвящённая 105-летию со дня рождения академика Л. В. Овсянникова

Y2 - 13 May 2024 through 17 May 2025

ER -