Asymptotic Growth and Decay of Two-Dimensional Symmetric Plasmas

TL;DR

This paper analyzes the long-term behavior of two-dimensional symmetric plasma models, revealing unbounded growth of particle trajectories, decay of electric fields, and the non-scattering nature of solutions due to persistent nonlinear interactions.

Contribution

It provides sharp asymptotic rates for particle growth, electric field decay, and charge density, and demonstrates the non-scattering behavior of solutions in two-dimensional plasma systems.

Findings

Particle positions and momenta grow unbounded as time increases.

Electric field decays at nearly sharp rates over time.

Solutions do not scatter to free-streaming profiles in 2D due to nonlinear effects.

Abstract

We study the large time behavior of classical solutions to the two-dimensional Vlasov-Poisson (VP) and relativistic Vlasov-Poisson (RVP) systems launched by radially-symmetric initial data with compact support. In particular, we prove that particle positions and momenta grow unbounded as $t \to \infty$ and obtain sharp rates on the maximal values of these quantities on the support of the distribution function for each system. Furthermore, we establish nearly sharp rates of decay for the associated electric field, as well as upper and lower bounds on the decay rate of the charge density in the large time limit. We prove that, unlike (VP) in higher dimensions, smooth solutions do not scatter to their free-streaming profiles as $t \to \infty$ because nonlinear, long-range field interactions dominate the behavior of characteristics due to the exchange of energy from the potential to the kinetic term. In this way, the system may "forget" any previous configuration of particles.