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 $${\mathbb {T}}^d$$ T d , $$d\ge 2$$ d ≥ 2 , via Newton’s second law through a supercritical mean-field limit. Namely, the coupling constant $$\lambda $$ λ in front of the pair potential, which is Coulombic, scales like $$N^{-\theta }$$ N - θ for some $$\theta \in (0,1)$$ θ ∈ ( 0 , 1 ) , in contrast to the usual mean-field scaling $$\lambda \sim N^{-1}$$ λ ∼ N - 1 . Assuming $$\theta \in (1-\frac{2}{d(d+1)},1)$$ θ ∈ ( 1 - 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 $$N\rightarrow \infty $$ N → ∞ . Han-Kwan and Iacobelli asked if their range for $$\theta $$ θ was optimal. We answer this question in the negative by showing the validity of the incompressible Euler equation in the limit $$N\rightarrow \infty $$ N → ∞ for $$\theta \in (1-\frac{2}{d},1)$$ θ ∈ ( 1 - 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 $$\theta $$ θ . Additionally, we show that for $$\theta \le 1-\frac{2}{d}$$ θ ≤ 1 - 2 d , one cannot, in general, expect convergence in the modulated energy notion of distance.