Arbitrary Polynomial Decay Rates of Neutral, Collisionless Plasmas

TL;DR

This paper demonstrates that solutions to the collisionless plasma modeled by the Vlasov-Poisson system can decay at arbitrarily fast polynomial rates, revealing an infinite set of asymptotic profiles and linear scattering behavior.

Contribution

It introduces the first construction of infinitely many asymptotic profiles with faster decay rates and establishes linear scattering results for all particle distributions.

Findings

Solutions can decay arbitrarily fast polynomially.

An infinite number of asymptotic profiles are characterized.

Particle distributions converge along characteristics at increasing rates.

Abstract

A multispecies, collisionless plasma is modeled by the Vlasov-Poisson system. Assuming the plasma is neutral and the electric field decays with sufficient rapidity as $t \to\infty$, we show that solutions can be constructed with arbitrarily fast, polynomial rates of decay, depending upon the properties of the limiting spatial average and its derivatives. In doing so, we establish, for the first time, a countably infinite number of asymptotic profiles for the charge density, electric field, and their derivatives, each of which is necessarily realized by a sufficiently smooth solution and exceeds the established dispersive decay rates. Finally, in each case we establish a linear $L^\infty$ scattering result for every particle distribution function, namely we show that they converge as $t \to \infty$ along the transported spatial characteristics at increasingly faster rates.