# Sharp asymptotic behavior of solutions of the $3d$ Vlasov-Maxwell system with small data

## TL;DR

This paper analyzes the long-term behavior of small data solutions to the 3D Vlasov-Maxwell system, achieving sharp decay estimates without compact support assumptions by leveraging vector field methods and null structures.

## Contribution

It introduces new vector field techniques and hierarchies to obtain optimal decay estimates for solutions of the 3D Vlasov-Maxwell system without restrictive initial data assumptions.

## Key findings

- Optimal decay rates for electromagnetic fields and derivatives.
- Control of high velocities through null structure.
- No need for compact support or neutrality assumptions.

## Abstract

We study the asymptotic properties of the small data solutions of the Vlasov-Maxwell system in dimension three. No neutral hypothesis nor compact support assumptions are made on the data. In particular, the initial decay in the velocity variable is optimal. We use vector field methods to obtain sharp pointwise decay estimates in null directions on the electromagnetic field and its derivatives. For the Vlasov field and its derivatives, we obtain optimal pointwise decay estimates by a vector field method where the commutators are modification of those of the free relativistic transport equation. In order to control high velocities and to deal with non integrable source terms, we make fundamental use of the null structure of the system and of several hierarchies in the commuted equations.