Strichartz estimates and moment bounds for the relativistic Vlasov-Maxwell system I. The $2$-D and $2\frac 12$-D cases

TL;DR

This paper establishes global existence, uniqueness, and regularity for the relativistic Vlasov-Maxwell system in 2D and 2.5D without compact momentum support, using Strichartz estimates and moment bounds.

Contribution

It extends prior results by removing the compact support assumption, allowing polynomial decay in momentum, and improves bounds on electromagnetic field growth.

Findings

Proves global solutions for non-compact initial data in 2D and 2.5D.

Uses Strichartz estimates to control solution moments.

Achieves improved bounds on electromagnetic field growth.

Abstract

Consider the relativistic Vlasov-Maxwell system with initial data of unrestricted size. In the two dimensional and the two and a half dimensional cases, Glassey-Schaeffer (1997, 1998, 1998) proved that for regular initial data with compact momentum support this system has unique global in time classical solutions. In this work we do not assume compact momentum support for the initial data and instead require only that the data have polynomial decay in momentum space. In the 2D and the $2\frac 12$D cases, we prove the global existence, uniqueness and regularity for solutions arising from this class of initial data. To this end we use Strichartz estimates and prove that suitable moments of the solution remain bounded. Moreover, we obtain a slight improvement of the temporal growth of the $L^\infty_x$ norms of the electromagnetic fields compared to Glassey-Schaeffer.