Derivation of Euler’s Equations of Perfect Fluids from von Neumann’s Equation with Magnetic Field

Abstract

We give a rigorous derivation of the incompressible 2D Euler equation from the von Neumann equation with an external magnetic field. The convergence is with respect to the modulated energy functional, and implies weak convergence in the sense of measures. This is the semi-classical counterpart of theorem 1.5 in (Han-Kwan and Iacobelli in Proc Am Math Soc 149(7):3045–3061, 2021). Our proof is based on a Gronwall estimate for the modulated energy functional, which in turn heavily relies on a recent functional inequality due to (Serfaty in Duke Math J 169:2887–2935, 2020).