Abstract: The Vlasov-Poisson model of boundless collisionless electron gas in self-consistent electric field continues to be one of the basic models for a number of modern physics areas, such as particle physics, electrodynamics, plasma physics, etc. This is due to simplicity, clarity, and obvious effectiveness of the model in describing complicated processes of the micro world. For example, the Vlasov-Poisson model is successfully used for development and operation of accelerators with colliding beams, which make it possible to accelerate elementary particles additionally by means of hot electron gas. In this report, the problem of linear stability for stationary solutions to the Vlasov-Poisson equations of electron plasma in one-dimensional formulation is investigated. The research objective is to prove an absolute linear instability for these solutions with respect to small one-dimensional perturbations. To achieve such goal, a transition from the Vlasov-Poisson equations to an infinite system of relations similar to the equations of isentropic flow of a compressible fluid medium in the “vortex shallow water” and the Boussinesq approximations was carried out. In the course of instability proof for stationary solutions to the Vlasov-Poisson equations in both kinetic and gas-dynamic descriptions, the well-known sufficient Newcomb-Gardner-Rosenbluth condition for stability of these solutions with regard to some incomplete unclosed subclasses of small one-dimensional perturbations was conversed. Also, the Gubarev inequalities were obtained for the Lyapunov functionals. The a priori exponential lower estimates for growth of small one-dimensional perturbations follow from these inequalities when the sufficient conditions for linear practical instability of the considered stationary solutions found in this report are satisfied. Since the obtained estimates were deduced without any additional restrictions on stationary solutions under study, then an absolute linear instability for these solutions with respect to small one-dimensional perturbations was thereby proved. The report results are confirmed by analytical examples of stationary solutions to the Vlasov-Poisson equations. Besides, they are fully consistent with the classical Earnshaw instability theorem and expand the area of applicability for this theorem from electrostatics to kinetics. Finally, constructiveness is inherent in the sufficient conditions for linear practical instability established here, which allows them to be used as a testing and control mechanism for conducting physical experiments and performing numerical calculations. In particular, these conditions can be applied to solve the problem of additional acceleration for elementary particles by electron plasma clumps. Specifically, they must be performed during the formation of a bunch of electrons. Due to this, it will take less time to create an electronic clot. On the contrary, after the electron cluster has formed, the conditions should not be fulfilled. Then the electronic clot will last longer. Thus, there is a real opportunity to control the acceleration of elementary particles with the help of electron plasma clumps.

Cite: Gubarev Y.G. , Zhang B.
Exploration of stability for stationary solutions to the Vlasov-Poisson equations of electron plasma in one-dimensional setting
In compilation Analytical and Numerical Methods in Differential Equations (Yanenko 105). A virtual conference on occasion of the 105th birthday of Academician Nikolai N. Yanenko. March 23-27, 2026. Book of Abstracts. – School of Mathematics and Geoinformatics. Suranaree University of Technology., 2026. – C.39.

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