On long-time behavior of monocharged and neutral plasma in one and one-half dimensions

Robert Glassey,Stephen Pankavich,Jack Schaeffer

doi:10.3934/krm.2009.2.465

Abstract

The motion of a collisionless plasma - a high-temperature, low-density, ionized gas - is described by the Vlasov-Maxwell system. In the presence of large velocities, relativistic corrections are meaningful, and when symmetry of the particle densities is assumed this formally becomes the relativistic Vlasov-Poisson system. These equations are considered in one space dimension and two momentum dimensions in both the monocharged (i.e., single species of ion) and neutral cases. The behavior of solutions to these systems is studied for large times, yielding estimates on the growth of particle momenta and a lower bound, uniform-in-time, on norms of the charge density. We also present similar results in the same dimensional settings for the classical Vlasov-Poisson system, which excludes relativistic effects.

Highlights

The fundamental equations which describe the time evolution of a collisionless plasma in the presence of large velocities are given by the relativistic Vlasov-Maxwell system (RVM)
The simplest form of these equations that retains electromagnetic effects can be obtained by posing the problem in one space dimension and two momentum dimensions, the so-called “one and one-half” dimensional system with a single species of ion:
Α which, by conservation of charge, guarantees ρ(t, x) dx = ρ(0, x) dx = 0 for all t ≥ 0. Another main focus of the paper will be to demonstrate the differences in qualitative behavior which arise when one compares monocharged and neutral plasmas, such as (RVP) with (RVPN) or (VP) when compared with (VPN). This is displayed by the differences in large time behavior of particle momenta for the systems (Theorems 1.3, 1.4, 1.5, and 1.6)

Summary

Introduction

The fundamental equations which describe the time evolution of a collisionless plasma in the presence of large velocities are given by the relativistic Vlasov-Maxwell system (RVM). This is displayed by the differences in large time behavior of particle momenta for the systems (Theorems 1.3, 1.4, 1.5, and 1.6). The only known results regarding long-time behavior for a relativistic plasma were obtained in [11] and, more recently, [8].

Results

Conclusion