On the Lagrangian structure of transport equations: Relativistic Vlasov systems

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 the existence of such solutions. 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 recently obtained for the Vlasov‐Poisson system in a series of papers by Ambrosio, Colombo, and Figalli; here, we analyze relativistic systems and also consider the contribution of the magnetic force into the evolution equation.