From microscopic to macroscopic scale equations: mean field, hydrodynamic and graph limits

We elaborate on various ways to pass to the limit a given family of finite-dimensional particle systems, either by mean field limit, deriving the Vlasov equation, or by hydrodynamic or graph limit, obtaining the Euler equation. We provide convergence estimates. We also show how to pass from Liouville to Vlasov or to Euler by taking adequate moments. Our results encompass and generalize a number of known results of the literature. As a surprising consequence of our analysis, we show that, under appropriate regularity assumptions, solutions of any quasilinear PDE can be approximated by the solutions of finite-dimensional particle systems