Global existence and modified scattering for the solutions to the Vlasov-Maxwell system with a small distribution function

TL;DR

This paper proves global existence and modified scattering for solutions to the Vlasov-Maxwell system with small initial data, using vector field methods and without support restrictions, and describes the asymptotic behavior of the electromagnetic field and distribution function.

Contribution

It provides a new proof of global existence without support restrictions and introduces a modified scattering framework with logarithmic corrections for the Vlasov-Maxwell system.

Findings

Electromagnetic field admits a radiation field along future null infinity.

Electromagnetic field approaches a smooth vacuum solution at large times.

Distribution function converges to a modified density with logarithmic corrections.

Abstract

The purpose of this paper is twofold. In the first part, we provide a new proof of the global existence of the solutions to the Vlasov-Maxwell system with a small initial distribution function. Our approach relies on vector field methods, together with the Glassey-Strauss decomposition of the electromagnetic field, and does not require any support restriction on the initial data or smallness assumption on the Maxwell field. Contrary to previous works on Vlasov systems in dimension $3$, we do not modify the linear commutators and avoid then many technical difficulties. In the second part of this paper, we prove a modified scattering result for these solutions. More precisely, we obtain that the electromagnetic field has a radiation field along future null infinity and approaches, for large time, a smooth solution to the vacuum Maxwell equations. As for the Vlasov-Poisson system, in constrast, the distribution function converges to a new density function along modifications of the characteristics of the free relativistic transport equation. In order to define these logarithmic corrections, we identify an effective asymptotic Lorentz force. By considering logarithmical modifications of the linear commutators, defined in terms of derivatives of the asymptotic Lorentz force, we finally prove higher order regularity results for the limit distribution function.