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 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 1.2 in [D. Han-Kwan and M. Iacobelli. Proc. Amer. 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 [S.

The magnetic Liouville equation as a semi-classical limit

“…The assumption that µ in is a Borel probability measure on R d × R d with finite second moments implies that the Toeplitz operator OP T ((2π ) d µ in ) belongs to the space D 2 (H)-see proposition (2.3) in [9]. This fact will also be restated explicitly in section (5).…”

“…Adapting the methods of [9] (especially theorem (4.1)), we formulate and prove a quantitative observation inequality for the Heisenberg equation (2), by utilizing the propagation estimate of section (3), as well as the optimal transport theory of section (5). The proof of the forthcoming coming theorem is almost identical to the proof in [9], and is included for the purpose of clarifying the link between theorem (2.4) and observability, which is not obvious-indeed the optimal transport approach demonstrated in [9] (which the same approach used here) is not a standard tool in some of the earlier literature on the subject .…”

“…However, we were are unable to locate a reference which includes a full treatment for the case d = 2, and therefore we decided to formulate theorem (2.4) in dimension ≥ 3, although likely this can be avoided with some more effort. Anyhow, the case of d = 2 with constant magnetic field of the type (A)-which is the case of interest for section (4)-has been already handled in lemma (6.6) in [5].…”

“…The main results are then proved in sections ( 3) and ( 4). How to deduce Monge-Kantorovich convergence from the evolution estimates of the latter sections is explained in section (5). Section ( 6) is an application of the main results to observation inequalities.…”