Abstract: This paper establishes a fundamental result on the stability of collisionless plasma: in a general case, the exact stationary solutions to multi-component Vlasov-Maxwell model are absolutely unstable with respect to small three-dimensional (3D) perturbations. Such conclusion determined by the direct Lyapunov method holds for a broad class of exact stationary solutions without imposing symmetrical or other simplifying restrictions. A key aspect of our approach is a hydrodynamic substitution that transforms the multi-component Vlasov-Maxwell equations into an infinite system of gas-dynamic equations, which is structurally analogous to mathematical models of compressible media. Within this framework, we derived an a priori exponential lower estimate for a subclass of growing in time small 3D perturbations, confirming the inherent instability for exact stationary solutions to the multi-component Vlasov-Maxwell model. Furthermore, we obtained sufficient conditions of constructive character for linear practical instability of the considered stationary solutions to multi-component Vlasov-Maxwell equations in relation to small 3D perturbations. Also, our findings significantly extend a scope of the Earnshaw theorem applicability from electrostatics to electromagnetism and kinetics. At last, this paper provides a new paradigm for plasma stability theory with straight implications for monitoring stability in controlled thermonuclear fusion research.

Cite: Gubarev Y.G. , Zhang Q.
On linear instability for three-dimensional dynamic equilibria of the multi-component Vlasov-Maxwell plasma
14th International Conference on Mathematical Modeling in Physical Sciences 20-23 Oct 2025