A general way to confined stationary Vlasov-Poisson plasma configurations

Abstract

We address the existence of stationary solutions of the Vlasov-Poisson system on a domain $\Omega \subset {\mathbb{R}}^3$ describing a high-temperature plasma which due to the influence of an external magnetic field is spatially confined to a subregion of $\Omega$. In a first part we provide such an existence result for a generalized system of Vlasov-Poisson type and investigate the relation between the strength of the external magnetic field, the sharpness of the confinement and the amount of plasma that is confined measured in terms of the total charges. The key tools here are the method of sub-/supersolutions and the use of first integrals in combination with cutoff functions. In a second part we apply these general results to the usual Vlasov-Poisson equation in three different settings: the infinite and finite cylinder, as well as domains with toroidal symmetry. This way we prove the existence of stationary solutions corresponding to a two-component plasma confined in a Mirror trap, as well as a Tokamak.

arXiv comment32 pages, 1 figure; in the updated version an appendix concerning a reference we cite has been added

References in corpus (5)

• Optimal control of a Vlasov-Poisson plasma by an external magnetic field
• Optimal control of a Vlasov-Poisson plasma by fixed magnetic field coils
• Confined steady states of a Vlasov-Poisson plasma in an infinitely long cylinder
• Confined Steady States of a Relativistic Vlasov-Maxwell Plasma in a Long Cylinder
• Optimal Control of a Hot Plasma

Cited by in corpus (2)

• Controlling a Vlasov-Poisson plasma by a Particle-In-Cell method based on a Monte Carlo framework
• On the two and one-half dimensional Vlasov-Poisson system with an external magnetic field: global well-posedness and stability of confined steady states