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

Abstract

A longstanding problem in mathematical physics is the rigorous derivation of the incompressible Euler equation from Newtonian mechanics. Recently, Han-Kwan and Iacobelli [16] showed that in the monokinetic regime, one can directly obtain the Euler equation from a system of N particles interacting in T d , d ≥ 2, via Newton's second law through a supercritical mean-field limit. Namely, the coupling constant λ in front of the pair potential, which is Coulombic, scales like N −θ for some θ ∈ (0, 1), in contrast to the usual mean-field scaling λ ∼ N −1 . Assuming 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 → ∞. Han-Kwan and Iacobelli asked if their range for θ was optimal. We answer this question in the negative by showing the validity of the incompressible Euler equation in the limit N → ∞ for θ ∈ (1 − 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 θ.

From Quantum Many-Body Systems to Ideal Fluids

“…). This is exactly the same range obtained by the author [Ros21] for the derivation of incompressible Euler from Newton's second law. Finally, we remark that our proof is, in fact, quantitative: we give an estimate (see Remark 2.3) for the differences ρ t ,ε,N :1 − 1 and J t ,ε,N :1 − u t in a negative-order fractional Sobolev spaces, which holds for arbitrary , ε, N .…”

“…If this is also the case in our dynamical setting, then we expect our scaling relation (1.7) to be sharp. Writing ε 2 N = N θ , we see that we can allow for 1 − 2 d < θ < 1, just as in the author's previous derivation of Euler from Newton [Ros21]. We leave the very interesting question of the effective limiting dynamics at and below the threshold θ = 1 − 2 d for future investigation.…”

“…The terminology coined by Han-Kwan and Iacobelli stems from the fact that the force term in (1.2) is more singular than in the usual mean-field regime, since it formally diverges as N → ∞. The author [Ros21] later extended the range of scalings for which the supercritical mean-field limit is valid to θ ∈ (1 − 2 d , 1) and gave arguments for the sharpness of this range in all dimensions. We also mention that the combined mean-field and quasineutral regime has been studied in [GPI18,GPI20b] without the monokinetic assumption, where the expected limiting equation is the so-called kinetic Euler equation, but only for very ε vanishing vary slowly as N → ∞.…”

“…Compared to the Golse-Paul functional, a key difference in our choice of (1.10) is the presence of the corrector ε 2 U t := ε 2 ∂ xα (u t ) β ∂ x β (u t ) α , with implicit summation over α, β, the choice of which is motivated by the aforementioned work of Han-Kwan and Iacobelli [HKI21] and the author [Ros21]. To understand the intuition behind the corrector, recall that the pressure p is obtained from the velocity field u through the Poisson equation −∆p = U, which follows by taking the divergence of both sides of (1.1) and using that u is divergence-free.…”

“…In some sense, the original functional used by Golse and Paul is the quantum analogue of the modulated energy used by Duerinckx and Serfaty [Ser20b,Appendix] to derive the pressureless Euler-Poisson equation from the Newton's second law with mean-field scaling in the monokinetic regime. Similarly, one may view our functional (1.10) as the quantum analogue of the modulated energy used by Han-Kwan and Iacobelli [HKI21] and the author [Ros21] to derive the incompressible Euler equation from Newton's second law with supercritical mean-field scaling in the monokinetic regime.…”