Overview
This document constitutes the second part of a two-paper series. Its primary objective is to establish the global existence of the 3D relativistic Vlasov-Maxwell (RVM) system for arbitrarily large, smooth, and localized initial data, specifically under conditions of cylindrical symmetry.
Part II focuses on developing two distinct sets of pointwise estimates for the localized electromagnetic field. These estimates are derived independently of the results presented in Part I. They leverage several characteristics of the system, including cylindrical symmetry, conservation laws, observed smoothing effects, and structural properties of the electromagnetic field, which encompass its null structure, double null structure, and the dichotomy of the magnetic field.
Research Context
The overall research program aims to demonstrate the global existence of the 3D RVM system for arbitrarily large smooth localized initial data, specifically when cylindrical symmetry is present. Part I of this two-paper sequence utilized an iterative smoothing scheme (ISS), a method inspired by the work of Klainerman and Staffilani (2002). Within the framework of Part I, the pointwise estimates developed in Part II were treated as axiomatic or 'black box' components.
Approach
The approach to establishing the pointwise estimates involves exploiting several inherent properties of the system. These include:
- Cylindrical symmetry.
- Conservation laws of the system.
- Smoothing effects identified within the system dynamics.
- Structural observations specific to the electromagnetic field, such as its null structure and double null structure.
- The magnetic field dichotomy.
A defining characteristic of these estimates is their assessment relative to the magnitude of the oscillation phase over a characteristic time. This method quantitatively captures the smoothing effect. Cases where no smoothing effect is observed are identified as error terms. Addressing these error terms constitutes the initial challenge in the bootstrap argument employed within the iterative smoothing scheme.
Findings
The research established two sets of pointwise estimates for the localized electromagnetic field:
- The first set provides a uniform upper bound across the localized electromagnetic field.
- The second set offers an upper bound that is contingent upon the distance from the z-axis. This second set yields improved control over the electromagnetic field when its location is further away from the z-axis.
These two sets of estimates are fundamental to the iterative smoothing scheme (ISS). The estimates effectively quantify smoothing effects by assessing them concerning the magnitude of the oscillation phase over characteristic time. From these estimates, it follows that scenarios devoid of any smoothing effect are categorized as error terms.