On the two and one-half dimensional Vlasov-Poisson system with an external magnetic field: global well-posedness and stability of confined steady states

TL;DR

This paper analyzes the two-and-a-half dimensional Vlasov-Poisson system with an external magnetic field, establishing global well-posedness, constructing confined steady states, and examining their stability under various perturbations.

Contribution

It provides the first rigorous analysis of stability for confined steady states in this specific plasma model with an external magnetic field.

Findings

Global well-posedness of the Cauchy problem established

Construction of steady states supported away from the confinement device

Stability of steady states analyzed via energy-Casimir method

Abstract

The time evolution of a two-component collisionless plasma is modeled by the Vlasov-Poisson system. In this work, the setting is two and one-half dimensional, that is, the distribution functions of the particles species are independent of the third space dimension. We consider the case that an external magnetic field is present in order to confine the plasma in a given infinitely long cylinder. After discussing global well-posedness of the corresponding Cauchy problem, we construct stationary solutions which indeed have support away from their confinement device. Then, in the main part of this work we investigate the stability of such steady states, both with respect to perturbations in the initial data, where we employ the energy-Casimir method, and also with respect to perturbations in the external magnetic field.