Abstract
In this talk, we explore the Lagrangian structure of relativistic Vlasov systems, including the relativistic Vlasov-Poisson equation and the quasi-electrostatic limit of the relativistic Vlasov-Maxwell equations. We present key results showing that renormalized solutions of these systems are Lagrangian, and that these solution concepts coincide. Consequently, we demonstrate that finite-energy solutions are transported by a global flow. Furthermore, we explore conditions for bounded energy, extending the notion of generalized solutions to "effective" densities. These results build on the work of Ambrosio, Colombo, and Figalli for Vlasov-Poisson systems, while incorporating relativistic effects and magnetic forces. Introductory material on relativistic Vlasov systems and related mathematical tools will be presented to ensure accessibility to a broad audience. This talk is based on joint work with Henrique Borrin (UNICAMP) and, hopefully, will be valuable for those interested in the analysis of partial differential equations in mathematical physics.
Brief Biography
Diego Marcon obtained his Ph.D. in Mathematics in 2013 from Instituto Superior Técnico (IST Lisbon, Portugal), as part of the UT Austin-Portugal CoLab Program. He held a postdoctoral position at the Federal University of Rio Grande do Sul (UFRGS) from 2013 to 2014, and since 2014, he has been an Assistant Professor at UFRGS. His research interests include partial differential equations, calculus of variations, optimal transport, and dynamical systems.
