On the Lagrangian structure of transport equations: relativistic Vlasov systems

TL;DR

This paper investigates the Lagrangian structure of relativistic Vlasov systems, demonstrating that solutions are transported by a global flow and extending existing results to include relativistic effects and magnetic forces.

Contribution

It establishes the Lagrangian nature of solutions for relativistic Vlasov systems and extends the theory to include magnetic forces and higher integrability conditions.

Findings

Solutions are Lagrangian and coincide with renormalized solutions.

Finite-energy solutions are transported by a global flow.

Existence of generalized solutions for effective densities.

Abstract

We study the Lagrangian structure of relativistic Vlasov systems, such as the relativistic Vlasov-Poisson and the relativistic quasi-eletrostatic limit of Vlasov-Maxwell equations. We show that renormalized solutions of these systems are Lagrangian and that these notions of solution, in fact, coincide. As a consequence, finite-energy solutions are shown to be transported by a global flow. Moreover, we extend the notion of generalized solution for "effective" densities and we prove its existence. Finally, under a higher integrability assumption of the initial condition, we show that solutions have every energy bounded, even in the gravitational case. These results extend to our setting those obtained by Ambrosio, Colombo, and Figalli \cite{vlasovpoisson} for the Vlasov-Poisson system; here, we analyse relativistic systems and we consider the contribution of the magnetic force into the evolution equation.