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

Abstract

We study the three dimensional many-particle quantum dynamics in meanfield 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. Contents 1. Introduction 1.1. Statement of the Main Theorem 1.2. Outline of the Proof 2. BBGKY Hierarchy v.s. H-NLS: Long-time Uniform in Estimates 2.1. A Tool Box of Space-time Estimates 2.2. A Klainerman-Machedon Bound 1st 2.3. Feeding the Strichartz Bound into the H 1 Estimate 2.4.