From Newton's second law to Euler's equations of perfect fluids

TL;DR

This paper rigorously derives the incompressible Euler equations from N-body Coulomb interactions using mean-field and singular limit analysis, advancing the mathematical understanding of fluid dynamics emergence.

Contribution

It introduces natural scalings and employs modulated energy methods to connect microscopic particle dynamics with macroscopic fluid equations.

Findings

Euler equations derived from Coulomb N-body dynamics

Valid in specific singular limits and scalings

Uses modulated energy methods for rigorous proof

Abstract

Vlasov equations can be formally derived from N-body dynamics in the mean-field limit. In some suitable singular limits, they may themselves converge to fluid dynamics equations. Motivated by this heuristic, we introduce natural scalings under which the incompressible Euler equations can be rigorously derived from N-body dynamics with repulsive Coulomb interaction. Our analysis is based on the modulated energy methods of Brenier and Serfaty.