Exact Large Time Behavior of Spherically-Symmetric Plasmas

TL;DR

This paper analyzes the long-term behavior of spherically symmetric plasmas described by the Vlasov-Poisson system, establishing optimal decay and growth rates, and characterizing the asymptotic distribution of particles.

Contribution

It provides the first comprehensive proof of optimal decay rates for all relevant norms and extends previous results to include sharp lower bounds and detailed asymptotic behavior.

Findings

Established optimal decay rates for charge density and electric field.

Proved sharp lower bounds and growth rates for particle support.

Showed convergence of spatial averages to a smooth, compactly-supported function.

Abstract

We consider the classical and relativistic Vlasov-Poisson systems with spherically-symmetric initial data and prove the optimal decay rates for all suitable $L^p$ norms of the charge density and electric field, as well as, the optimal growth rates for the largest particle position and momentum on the support of the distribution function. Though a previous work \cite{Horst} established upper bounds on the decay of the supremum of the charge density and electric field, we provide a slightly different proof, attain optimal rates, and extend this result to include all other norms. Additionally, we prove sharp lower bounds on each of the aforementioned quantities and establish the time-asymptotic behavior of all spatial and momentum characteristics. Finally, we investigate the limiting behavior of the spatial average of the particle distribution as $t \to \infty$. In particular, we show that it converges uniformly to a smooth, compactly-supported function that preserves the mass, angular momentum, and energy of the system and depends only upon limiting particle momenta.