Instability of Nonmonotone Magnetic Equilibria of the Relativistic Vlasov-Maxwell System

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.Comment: 46 page

On the spectrum of shear flows and uniform ergodic theorems

The spectra of parallel flows (that is, flows governed by first-order differential operators parallel to one direction) are investigated, on both $L^2$ spaces and weighted-$L^2$ spaces. As a consequence, an example of a flow admitting a purely singular continuous spectrum is provided. For flows admitting more regular spectra the density of states is analyzed, and spaces on which it is uniformly bounded are identified. As an application, an ergodic theorem with uniform convergence is proved.Comment: 18 pages, no figure

Averaging along degenerate flows on the annulus

Periodic flows on the annulus are considered. For flows that degenerate (i.e. the flow becomes arbitrarily slow along some flow lines) a convergence rate for the averaging of functions along the flow is obtained. This rate -- which is slower than the rate obtained for flows that do not degenerate (i.e. when there is a spectral gap) -- holds on an appropriate functional subspace. The main ingredient is an estimate of the density of the spectrum of the generator near zero.Comment: 15 pages. All comments are welcom