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

TL;DR

This paper establishes the equivalence of Lagrangian and Eulerian descriptions for the Vlasov-Poisson system with specular reflection in smooth domains, proving existence of renormalized solutions and global flows in dimensions 3 and 4.

Contribution

It extends recent theoretical frameworks to domains with boundary, demonstrating the existence and properties of solutions and flows under specular reflection conditions.

Findings

Equivalence of Lagrangian and Eulerian descriptions in bounded domains.

Existence of renormalized solutions with bounded energy.

Global flows established for dimensions 3 and 4.

Abstract

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 $d = 3, 4$.