Quantum Mean-field Limit to the Compressible Fluids

Abstract

We study the three dimensional quantum many-body dynamics with a delta-type potential N^3β V(N ^β x) and a Coulomb potential. As the particle number N tends to infinity and the Planck’s constant ħ tends to zero independently, we prove the weak convergence of the quantum mass and momentum densities to the Euler–Poisson equation with the pressure before its blow-up time. The proof is based on the modulated energy method in the setting of the quantum many-body dynamics, for which the key is a quantum functional inequality according to the interaction potentials. In Golse–Paul [Comm. Pure Appl. Math., 2022, 75(6): 1332–1376], the functional inequality for the Coulomb potential is achieved based on Serfaty’s inequality [Duke Math. J., 2020, 169(15): 2887–2935]. In the mean-field regime

\beta \in \left(0,\frac{1}{3}\right)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\beta \in \left({0,{1 \over 3}} \right)$$\end{document}

, we prove the quantum functional inequality for the delta-type potential under technical profile assumptions on the potential V(x).

Keywords

Euler–Poisson equations / quantum many-body dynamics / modulated energy / functional inequality

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