On the rigorous derivation of the incompressible Euler equation from Newton’s second law

Abstract

A long-standing problem in mathematical physics is the rigorous derivation of the incompressible Euler equation from Newtonian mechanics. Recently, Han-Kwan and Iacobelli (Proc Am Math Soc 149:3045–3061, 2021) showed that in the monokinetic regime, one can directly obtain the Euler equation from a system of N particles interacting in {mathbb {T}}^d, dge 2, via Newton’s second law through a supercritical mean-field limit. Namely, the coupling constant lambda in front of the pair potential, which is Coulombic, scales like N^{-theta } for some theta in (0,1), in contrast to the usual mean-field scaling lambda sim N^{-1}. Assuming theta in (1-frac{2}{d(d+1)},1), they showed that the empirical measure of the system is effectively described by the solution to the Euler equation as Nrightarrow infty . Han-Kwan and Iacobelli asked if their range for theta was optimal. We answer this question in the negative by showing the validity of the incompressible Euler equation in the limit Nrightarrow infty for theta in (1-frac{2}{d},1). Our proof is based on Serfaty’s modulated-energy method, but compared to that of Han-Kwan and Iacobelli, crucially uses an improved “renormalized commutator” estimate to obtain the larger range for theta . Additionally, we show that for theta le 1-frac{2}{d}, one cannot, in general, expect convergence in the modulated energy notion of distance.