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

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

Сборник

Analytical Methods in Differential Equations
Сборник, De Gruyter. Berlin, Boston.2025. 218 c. ISBN 9783111570518.

Вых. Данные

Год: 2025, Страницы: 111-120 Страниц : 10 DOI: 10.1515/978311157058-012

Ключевые слова

Vlasov–Poisson equations, Stationary solutions, Small perturbations, Direct Lyapunov method, Instability

Авторы

Gubarev Yuriy ^1,2 , Liu Yang ²

Организации

1	Laboratory for Explosion Physics, Lavrentyev Institute for Hydrodynamics, 15 Lavrentyev Av., Novosibirsk, 630090, Russian Federation
2	Department for Differential Equations, Novosibirsk State University, 1 Pirogova Str., Novosibirsk, 630090, Russian Federation

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.

Gubarev Y. , Liu Y.
On the instability for one subclass of three-dimensional dynamic equilibrium states of the electron Vlasov–Poisson gas
В сборнике Analytical Methods in Differential Equations. – De Gruyter., 2025. – C.111-120. – ISBN 9783111570518. DOI: 10.1515/978311157058-012

Опубликована в печати:	17 февр. 2025 г.
Опубликована online:	17 февр. 2025 г.

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