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

Understand

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 $N\rightarrow\infty$.