The Lagrangian structure of the Vlasov-Poisson system in domains with specular reflection

In this work, we deal with the Vlasov-Poisson system in smooth physical domains with specular boundary condition, under mild integrability assumptions, and $d \ge 3$. We show that the Lagrangian and Eulerian descriptions of the system are also equivalent in this context by extending the recent developments by Ambrosio, Colombo, and Figalli to our setting. In particular, assuming that the total energy is bounded, we prove the existence of renormalized solutions, and we also show that they are transported by a weak notion of flow that allows velocity jumps at the boundary. Finally, we show that flows can be globally defined for $d = 3, 4$.