Modified Scattering of Solutions to the Relativistic Vlasov-Maxwell System Inside the Light Cone

TL;DR

This paper analyzes the long-time behavior of solutions to the relativistic Vlasov-Maxwell system, showing they scatter to a modified free-streaming profile with detailed asymptotics for fields and densities.

Contribution

It establishes the precise asymptotic behavior and scattering results for solutions with small, compactly supported initial data, including the construction of limiting fields and modified scattering.

Findings

Solutions scatter to a modified free-streaming profile.

The spatial average of the particle distribution converges to a smooth, compactly-supported limit.

Decay rates depend on the limiting charge density, with sharper estimates when it vanishes.

Abstract

We consider the relativistic Vlasov-Maxwell system in three dimensions and study the limiting asymptotic behavior as $t \to \infty$ of solutions launched by small, compactly supported initial data. In particular, we prove that such solutions scatter to a modification of the free-streaming asymptotic profile. More specifically, we show that the spatial average of the particle distribution function converges to a smooth, compactly-supported limit and establish the precise, self-similar asymptotic behavior of the electric and magnetic fields, as well as, the macroscopic densities and their derivatives in terms of this limiting function. Upon constructing the limiting fields, a modified $L^\infty$ scattering result for the particle distribution function along the associated trajectories of free transport corrected by the limiting Lorentz force is then obtained. When the limiting charge density does not vanish, our estimates are sharp up to a logarithmic correction. However, when this quantity is identically zero in the limit, the limiting current density and fields may also vanish, which gives rise to decay rates that are faster than those attributed to the dispersive mechanisms in the system.