APDEs Seminar

This talk will provide an overview of intermediate interactions in particle systems and their applications to fluid dynamics and biological modeling. The discussion will begin with an introduction to scaling limits, highlighting the distinctions between Mean Field, local, and intermediate interaction cases, which serve as bridges between microscopic particle dynamics and macroscopic PDE formulations. Next, I will explore two applications of this type of interaction: a microscopic approach to the Vlasov-Fokker-Planck-Navier-Stokes equations, which model particle-fluid interactions, and a PDE model for cell-cell adhesion, with a focus on biological aggregation phenomena. The main results will illustrate the convergence properties of empirical measures, achieved through energy estimates and tightness conditions, while addressing challenges in particle-fluid coupling and stochastic modeling. Finally, I will conclude with a discussion of open questions, particularly those related to the problem of fluctuations.