Instability of nonmonotone magnetic equilibria of the relativistic Vlasov–Maxwell system

Abstract

We consider the question of linear instability of an equilibrium of the Relativistic Vlasov-Maxwell (RVM) System that has a strong magnetic field. Standard instability results deal with systems where there are fewer particles with higher energies. In this paper we extend those results to the class of equilibria for which the number of particles does not depend monotonically on the energy. Without the standard sign assumptions, the analysis becomes significantly more involved.

Instabilities of the Relativistic Vlasov--Maxwell System on Unbounded Domains

“…Two notable later results are [6,11]. We refer to [1] for additional references. The current result continues a program initiated by Lin and Strauss [9,10] and continued by the first author [1,2].…”

“…is the linearised Vlasov transport operator. We then invert (1.13) by applying λ (λ + D) −1 , which is an ergodic averaging operator along the trajectories of D (depending upon λ as a parameter), see [1,Eq. (2.10)].…”

“…We refer to [1] for additional references. The current result continues a program initiated by Lin and Strauss [9,10] and continued by the first author [1,2]. In [9,10] the equilibria were always assumed to be strongly monotone, in the sense of (1.7).…”

“…Remark 3.1. Operators Q λ ± also appeared in the prior works [9,10,1,2]. In each of these Q λ ± were defined as an integrated average over the characteristics of the operators D ± .…”

Instabilities in Kinetic Theory and Their Relationship to the Ergodic Theorem

“…There are two parallel approaches for inverting this equation in order to obtain an expression for f ± . In the first, which can be found in [1], we integrate (2.3) along the trajectories (X ± (s; x, v), V ± (s; x, v)) of the vectorfields D ± in phase space, which satisfẏ…”