Quantum Quasi-neutral Limits and Isothermal Euler Equations
Immanuel Ben Porat, Gui-Qiang G. Chen, Difan Yuan
TL;DR
This work provides a rigorous derivation of the isothermal Euler equations from quantum and semiclassical models, via combined quasi-neutral and mean-field limits of the Schrödinger-Poisson-Boltzmann system and its 1D N-body von Neumann counterpart. The authors employ modulated energy techniques to obtain stability and convergence results, including a Grönwall-type bound for the 1-body problem and Serfaty-based commutator estimates for the N-body case, all while handling the exponential nonlinearity in the Poisson-Boltzmann coupling. They establish well-posedness of the underlying quantum dynamics and construct well-prepared initial data that drive the macroscopic limits, thereby linking quantum microscopic dynamics to macroscopic quantum hydrodynamic models. The results advance the mathematical understanding of quantum-classical interactions in compressible fluids and provide a blueprint for deriving quantum hydrodynamic limits from many-body quantum systems.
Abstract
We provide a rigorous justification of the semiclassical quasi-neutral and the quantum many-body limits to the isothermal Euler equations. We consider the nonlinear Schrödinger-Poisson-Boltzmann system under a quasi-neutral scaling and establish the convergence of its solutions to the isothermal Euler equations. Different from the previous results that dealt with the linear Poisson equations, the system under our consideration accounts for the exponential nonlinearity in the potential. A modulated energy method is adopted, allowing us to derive the stability estimates and asymptotics. Furthermore, we focus our analysis on the many-body quantum problem via the von Neumann equation and establish a mean-field limit in one dimension by using Serfaty's functional inequalities, and thus connecting the quantum many-body dynamics with the macroscopic hydrodynamic equations. A refined analysis of the quasi-neutral scaling for the massless systems is presented, and the well-posedness of the underlying quantum dynamics is established. Moreover, the construction of general admissible initial data is obtained. Our results provide a rigorous mathematical analysis for the derivation of quantum hydrodynamic models and their limits, contributing to the broader understanding of interactions between quantum mechanics and compressible fluid dynamics.