Landau damping in relativistic plasmas

We examine the phenomenon of Landau damping in relativistic plasmas via a study of the relativistic Vlasov-Poisson (rVP) system on the torus for initial data sufficiently close to a spatially uniform steady state. We find that if the steady state is regular enough (essentially in a Gevrey class of degree in a specified range) and if the deviation of the initial data from this steady state is small enough in a certain norm, the evolution of the system is such that its spatial density approaches a uniform constant value quasi-exponentially fast (i.e., like exp(-C vertical bar t vertical bar((nu) over bar)) for (nu) over bar is an element of (0,1)). We take as a priori assumptions that solutions launched by such initial data exist for all times (by no means guaranteed with rVP, but a reasonable assumption since we are close to a spatially uniform state) and that the various norms in question are continuous in time (which should be a consequence of an abstract version of the Cauchy-Kovalevskaya theorem). In addition, we must assume a kind of "reverse Poincare inequality" on the Fourier transform of the solution. In spirit, this assumption amounts to the requirement that there exists 0 < x < 1 so that the mass in the annulus x <= vertical bar upsilon vertical bar < 1 for the solution launched by the initial data is uniformly small for all t. Typical velocity bounds for solutions to rVP launched by small initial data (at least on R-6) imply this bound. We note that none of our results require spherical symmetry (a crucial assumption for many current results on rVP). (C) 2016 AIP Publishing LLC.