Hydrodynamic limits of the nonlinear Schr\"odinger equation with the Chern-Simons gauge fields

TL;DR

This paper investigates the hydrodynamic limits of the nonlinear Schr"odinger-Chern-Simons system under different scalings, demonstrating convergence to classical fluid equations coupled with gauge fields as quantum effects vanish.

Contribution

It introduces two new scalings of the Schr"odinger-Chern-Simons system and proves their convergence to Euler-type equations using modulated energy estimates.

Findings

Convergence to compressible Euler system with quantum correction.

Convergence to incompressible Euler system with gauge coupling.

Application of relative entropy method in quantum hydrodynamics.

Abstract

We present two types of the hydrodynamic limit of the nonlinear Schr\"odinger-Chern-Simons (SCS) system. We consider two different scalings of the SCS system and show that each SCS system asymptotically converges towards the compressible and incompressible Euler system, coupled with the Chern-Simons equations and Poisson equation respectively, as the scaled Planck constant converges to 0. Our method is based on the modulated energy estimate. In the case of compressible limit, we observe that the classical theory of relative entropy method can be applied to show the hydrodynamic limit, with the additional quantum correction term. On the other hand, for the incompressible limit, we directly estimate the modulated energy to derive the desired asymptotic convergence.