Regular Lagrangian flow for wavelike vector fields and the Vlasov-Maxwell system

In this paper, we study the Lagrangian structure of Vlasov-Maxwell system, that is, by using a suitable notion of flow, we prove that if the densities $\rho ,j$ are integrable in spacetime, and the charge acceleration ${\partial }_{t}j$ and ${\partial }_{tt}j$ (or $\nabla {\partial }_{t}j$) are integrable functions in spacetime, then renormalized and distributional solutions of the system are the transport of the initial condition by its flow. We study more general vector fields, with wavelike structure in the sense that it has finite speed of propagation, generalizing the vector fields studied in [6]. The result is a extension of those obtained by Ambrosio, Colombo, and Figalli [2] for the Vlasov-Poisson system, and by the author and Marcon [5] for relativistic Vlasov-systems with quasistatic approximations of Maxwell's equations.