Landau Damping in Relativistic Plasmas

Abstract

We examine the phenomenon of Landau Damping in relativistic plasmas via a study of the relativistic Vlasov-Poisson system (rVP) 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 that 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 sub-exponentially fast (i.e. like $\exp (-C|t{|}^{\overline{\nu }})$ for $\overline{\nu }\in (0,1)$). We take as \emph{a priori} assumptions that solutions launched by such initial data exist for all times (by no means guaranteed with rVP, but reasonable 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 Poincaré inequality" on the Fourier transform of the solution. In spirit, this assumption amounts to the requirement that there exists $0<\varkappa <1$ so that the mass in the annulus $\varkappa \le |v|<1$ for the solution launched by the initial data is uniformly small for all $t$.

88 pages, 3 figures

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