2021/04/23 by Matthew Rosenzweig, Rosenzweig, Matthew · 1 citation
Engineering · Environmental Science · #Fluid Dynamics and Turbulent Flows #Climate variability and models #Phase Equilibria and Thermodynamics
paper · pdf · doi:10.48550/arxiv.2104.11723
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 \mathbbTd, 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.