Abstract: In the electrostatic approximation, when the electric field of electrons and ions is self-consistent, the plasma dynamics is described by the kinetic Vlasov-Poisson equations. In this case, such equations are used to study the collisionless motion of electrons, which interact with each other through the Coulomb repulsive forces, against the background of a homogeneous distribution of ions in the whole physical continuum. The aim of this research is to prove an absolute instability for the exact stationary cylindrically symmetrical solutions to kinetic Vlasov-Poisson equations by the direct Lyapunov method with respect to the small cylindrically symmetrical perturbations. The results of this study are important for solving the problem of controlled thermonuclear fusion. To achieve such goal, the hydrodynamic substitution of independent variables is performed so that kinetic Vlasov-Poisson equations are transformed to an infinite system of cylindrically symmetrical equations similar to the equations of isentropic flows of compressible fluid medium in the vortex shallow water and the Boussinesq approximations. The new defining equations have the exact stationary solutions that are equivalent to the exact stationary cylindrically symmetrical solutions to kinetic Vlasov-Poisson equations. Then these defining 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 new defining equations, which grow over time and are described by the field of Lagrangian displacements. Since the estimate is obtained for any exact stationary solutions to new defining equations, it proves precisely the absolute linear instability of these solutions with regard to the small cylindrically symmetrical perturbations from the subclass mentioned above. Thus, the Newcomb-Gardner-Rosenbluth sufficient condition for linear stability of the exact stationary cylindrically symmetrical solutions to kinetic Vlasov-Poisson equations is conversed, and its formal character is revealed. Also, the sufficient conditions for linear practical instability of the exact stationary solutions to new defining equations are found, and their constructive nature is discovered. At last, the results of this research are consistent with the well-known Earnshaw theorem on instability in electrostatics and extend the scope of its applicability from classical mechanics to statistical one. As for the significance of these results, they can be used to study the adequacy of mathematical models for plasma to the physical phenomena which the models describe. Furthermore, the results obtained here can be applied to the development and subsequent operation of devices designed to perform the controlled thermonuclear fusion. In order for a plasma confinement device to operate reliably, it needs for us to ensure the practical stability of its dynamic equilibrium states with respect to all acceptable perturbations. In particular, these equilibrium states should be robust in a practical sense for small cylindrically symmetrical perturbations. This can be achieved by creation of numerical and physical models, which correspond to the linearized initial-boundary value problem under investigation, with control the sufficient conditions for linear practical instability at some reference time points. In constructing these models, the main focus should be on ensuring that the sufficient conditions for linear practical instability are not met at the expense of those or other known external influences on small cylindrically symmetrical perturbations growing with time (for example, by virtue of suitable setting of initial conditions). In consequence, the operation reliability of the device for plasma confinement in working mode will be guaranteed.

Cite: Gubarev Y.G. , Luo J.
Study of linear stability for cylindrically symmetrical states of dynamic equilibrium of two-component Vlasov-Poisson plasma
Russian-Chinese Conference “Differential and Difference Equations” 02-06 нояб. 2023