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

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 ofNparticles interacting in$${\mathbb {T}}^d$$Td,$$d\ge 2$$d2, via Newton’s second law through asupercritical mean-field limit. Namely, the coupling constant$$\lambda $$λin front of the pair potential, which is Coulombic, scales like$$N^{-\theta }$$N-θfor some$$\theta \in (0,1)$$θ(0,1), in contrast to the usual mean-field scaling$$\lambda \sim N^{-1}$$λN-1. Assuming$$\theta \in (1-\frac{2}{d(d+1)},1)$$θ(1-2d(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$$\theta $$θwas optimal. We answer this question in the negative by showing the validity of the incompressible Euler equation in the limit$$N\rightarrow \infty $$Nfor$$\theta \in (1-\frac{2}{d},1)$$θ(1-2d,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$$\theta \le 1-\frac{2}{d}$$θ1-2d, one cannot, in general, expect convergence in the modulated energy notion of distance.

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Author(s) / Creator(s):
Publisher / Repository:
Springer Science + Business Media
Date Published:
Journal Name:
Letters in Mathematical Physics
Medium: X
Sponsoring Org:
National Science Foundation
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