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

TL;DR

This paper extends the rigorous derivation of the incompressible Euler equation from Newtonian particle systems, showing it holds for a broader range of interaction scaling parameters than previously established.

Contribution

It proves the Euler equation's validity in the mean-field limit for a larger range of the coupling constant scaling parameter, improving upon prior results by using an enhanced analytical estimate.

Findings

Validates Euler equation for   (1 - 2/d) < heta < 1 in all dimensions

Uses Serfaty's modulated-energy method with improved estimates

Extends the known range of  for deriving Euler from particle systems

Abstract

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 $\mathbb{T}^d$, $d\geq 2$, 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 $N\rightarrow\infty$. 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 $N\rightarrow\infty$ 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 $\theta$.