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

Abstract

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.