The derivation of the compressible Euler equation from quantum many-body dynamics

TL;DR

This paper rigorously derives the compressible Euler equation from three-dimensional quantum many-body dynamics in the mean-field limit, linking microscopic quantum interactions to macroscopic fluid behavior.

Contribution

It combines hierarchy and modulated energy methods to establish the microscopic to macroscopic convergence and provides a physical interpretation of Strichartz bounds.

Findings

Proves convergence of quantum dynamics to Euler equations as particle number and Planck's constant tend to limits.

Shows pressure emerges from microscopic interactions via space-time averages.

Establishes bounds up to the first blow-up time of the Euler solution.

Abstract

We study the three dimensional many-particle quantum dynamics in mean-field setting. We forge together the hierarchy method and the modulated energy method. We prove rigorously that the compressible Euler equation is the limit as the particle number tends to infinity and the Planck's constant tends to zero. We establish strong and quantitative microscopic to macroscopic convergence of mass and momentum densities up to the 1st blow up time of the limiting Euler equation. We justify that the macroscopic pressure emerges from the space-time averages of microscopic interactions, which are in fact, Strichartz-type bounds. We have hence found a physical meaning for Strichartz type bounds which were first raised by Klainerman and Machedon in this context.