On the Rigorous Derivation of the Incompressible Euler Equation from Newton's Second Law

A longstanding problem in mathematical physics is the rigorous derivation of the incompressible Euler equation from Newtonian mechanics. Recently, Han-Kwan and Iacobelli arXiv:2006.14924 showed that in the monokinetic regime, one can directly obtain the Euler equation from a system of $N$ particles interacting in ${\mathbb{T}}^d$, $d\ge 2$, 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 $\theta \in (0,1)$,incontrasttotheusualmean-fieldscaling$\lambda\sim N^{-1}$. 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 $N\to \mathrm{\infty }$. 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 $N\to \mathrm{\infty }$ for $\theta \in (1-\frac{2}{d},1)$. For reasons of scaling, this range appears optimal in all dimensions. 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 $\theta$.