# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021202
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## Hydrodynamic limits of the nonlinear Schrödinger equation with the Chern-Simons gauge fields

 Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea

*Corresponding author: Bora Moon

Received  May 2021 Revised  October 2021 Early access January 2022

Fund Project: The work of J. Kim and B. Moon was supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science and ICT (NRF-2020R1A4A3079066) and the work of B. Moon was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2019R1I1A1A01059585)

We present two types of the hydrodynamic limit of the nonlinear Schrödinger-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.

Citation: Jeongho Kim, Bora Moon. Hydrodynamic limits of the nonlinear Schrödinger equation with the Chern-Simons gauge fields. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021202
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