# American Institute of Mathematical Sciences

June  2022, 42(6): 2541-2561. doi: 10.3934/dcds.2021202

## 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 Published  June 2022 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 and Continuous Dynamical Systems, 2022, 42 (6) : 2541-2561. doi: 10.3934/dcds.2021202
##### References:
 [1] C. Bardos, F. Golse and C. D. Levermore, The acoustic limit for the Boltzmann equation, Arch. Ration. Mech. Anal., 153 (2000), 177-204.  doi: 10.1007/s002050000080. [2] L. Bergé, A. de Bouard and J.-C. Saut, Blowing up time-dependent solutions of the planar, Chern-Simons gauged nonlinear Schrödinger equation, Nonlinearity, 8 (1995), 235-253.  doi: 10.1088/0951-7715/8/2/007. [3] F. Bouchut, F. Golse and M. Pulvirenti, Kinetic Equations and Asymptotic Theory, Series in Applied Mathematics. Gauthier–Villars, 2000. [4] C. M. Dafermos, The second law of thermodynamics and stability, Arch. Rational Mech. Anal., 70 (1979), 167-179.  doi: 10.1007/BF00250353. [5] Z. F. Ezawa, M. Hotta and A. Iwazaki, Breathing vortex solitons in nonrelativistic Chern-Simons gauge theory, Phys. Rev. Lett., 67 (1991), 411-414.  doi: 10.1103/PhysRevLett.67.411. [6] F. Golse and L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math., 155 (2004), 81-161.  doi: 10.1007/s00222-003-0316-5. [7] P. Goncalves, C. Landim and C. Tonielli, Hydrodynamic limit for a particle system with degenerate rates, Ann. Inst. Henri Poincaré Probab. Stat., 45 (2009), 887-909.  doi: 10.1214/09-AIHP210. [8] T. Goudin, P.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. II Fine particles regime, Indiana Univ. Math. J., 53 (2004), 1517-1536.  doi: 10.1512/iumj.2004.53.2509. [9] H. Huh, Energy solution to the Chern-Simons-Schrödinger equations, Abstr. Appl. Anal., 2013 (2013), Article ID 590653. doi: 10.1155/2013/590653. [10] R. Jackiw and S.-Y. Pi, Classical and quantal nonrelativistic Chern-Simons theory, Phys. Rev. D, 42 (1990), 3500-3513.  doi: 10.1103/PhysRevD.42.3500. [11] R. Jackiw and S.-Y. Pi, Soliton solutions to the gauged nonlinear Schrödinger equation on the plane, Phys. Rev. Lett., 64 (1990), 2969-2972.  doi: 10.1103/PhysRevLett.64.2969. [12] M.-J. Kang and A. Vasseur, Asymptotic analysis of Vlasov-type equations under strong local alignment regime, Math. Models Methods Appl. Sci., 25 (2015), 2153-2173.  doi: 10.1142/S0218202515500542. [13] C. Landim, Hydrodynamic limit of interacting particle systems,, School and Conference on Probability Theory, ICTP Lect. Notes, XVII, Abdus Salam Int. Cent. Theoret. Phys., Trieste, (2004), 57–100. [14] C. C. Lee and T. C. Lin, Incompressible and compressible limits of two-component Gross-Pitaevskii equations with rotating fields and trap potentials, J. Math. Phys., 49 (2008), 043517.  doi: 10.1063/1.2912716. [15] H. L. Li and C. K. Lin, Zero Debye length asymptotic of the quantum hydrodynamic model of semiconductors, Comm. Math. Phys., 256 (2005), 195-212.  doi: 10.1007/s00220-005-1316-7. [16] Z. M. Lim, Large data well-posedness in the energy space of the Chern-Simons-Schrödinger system, J. Differential Equations, 264 (2018), 2553-2597.  doi: 10.1016/j.jde.2017.10.026. [17] P.-L. Lions, Mathematical Topics in Fluid Mechanics, The Clarendon Press, Oxford University Press, New York, 1998. [18] P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl., 77 (1998), 585-627.  doi: 10.1016/S0021-7824(98)80139-6. [19] B. Liu and P. Smith, Global wellposedness of the equivariant Chern-Simons-Schrödinger equation, Rev. Mat. Iberoam., 32 (2016), 751-794.  doi: 10.4171/RMI/898. [20] B. Liu, P. Smith and D. Tataru, Local wellposedness of Chern-Simons-Schrödinger equation, Int. Math. Res. Not. IMRN, 2014 (2014), 6341-6398.  doi: 10.1093/imrn/rnt161. [21] C.-K. Lin and K.-C. Wu, Hydrodynamic limits of the nonlinear Klein-Gordon equation, J. Math. Pures Appl., 98 (2012), 328-345.  doi: 10.1016/j.matpur.2012.02.002. [22] T. C. Lin and P. Zhang, Incompressible and compressible limits of coupled systems of nonlinear Schrödinger equations, Commun. Math. Phys., 266 (2006), 547-569.  doi: 10.1007/s00220-006-0061-x. [23] E. Madelung, Quantentheorie in hydrodynamischer form, Z. Physik, 40 (1927), 322. [24] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7. [25] A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge Univ. Press. 2002. [26] A. Mellet and A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations, Commun. Math. Phys., 281 (2008), 573-596.  doi: 10.1007/s00220-008-0523-4. [27] M. Puel, Convergence of the Schrödinger-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 27 (2002), 2311-2331.  doi: 10.1081/PDE-120016159. [28] L. Saint-Raymond, Convergence of solutions to the Boltzmann equation in the incompressible Euler limit, Arch. Rational Mech. Anal., 166 (2003), 47-80.  doi: 10.1007/s00205-002-0228-3. [29] L. Saint-Raymond, Hydrodynamic limits: Some improvements of the relative entropy methods, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 705-744.  doi: 10.1016/j.anihpc.2008.01.001. [30] H. T. Yau, Relative entropy and hydrodynamics of Ginzberg-Landau models, Lett. Math. Phys., 22 (1991), 63-80.  doi: 10.1007/BF00400379.

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##### References:
 [1] C. Bardos, F. Golse and C. D. Levermore, The acoustic limit for the Boltzmann equation, Arch. Ration. Mech. Anal., 153 (2000), 177-204.  doi: 10.1007/s002050000080. [2] L. Bergé, A. de Bouard and J.-C. Saut, Blowing up time-dependent solutions of the planar, Chern-Simons gauged nonlinear Schrödinger equation, Nonlinearity, 8 (1995), 235-253.  doi: 10.1088/0951-7715/8/2/007. [3] F. Bouchut, F. Golse and M. Pulvirenti, Kinetic Equations and Asymptotic Theory, Series in Applied Mathematics. Gauthier–Villars, 2000. [4] C. M. Dafermos, The second law of thermodynamics and stability, Arch. Rational Mech. Anal., 70 (1979), 167-179.  doi: 10.1007/BF00250353. [5] Z. F. Ezawa, M. Hotta and A. Iwazaki, Breathing vortex solitons in nonrelativistic Chern-Simons gauge theory, Phys. Rev. Lett., 67 (1991), 411-414.  doi: 10.1103/PhysRevLett.67.411. [6] F. Golse and L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math., 155 (2004), 81-161.  doi: 10.1007/s00222-003-0316-5. [7] P. Goncalves, C. Landim and C. Tonielli, Hydrodynamic limit for a particle system with degenerate rates, Ann. Inst. Henri Poincaré Probab. Stat., 45 (2009), 887-909.  doi: 10.1214/09-AIHP210. [8] T. Goudin, P.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. II Fine particles regime, Indiana Univ. Math. J., 53 (2004), 1517-1536.  doi: 10.1512/iumj.2004.53.2509. [9] H. Huh, Energy solution to the Chern-Simons-Schrödinger equations, Abstr. Appl. Anal., 2013 (2013), Article ID 590653. doi: 10.1155/2013/590653. [10] R. Jackiw and S.-Y. Pi, Classical and quantal nonrelativistic Chern-Simons theory, Phys. Rev. D, 42 (1990), 3500-3513.  doi: 10.1103/PhysRevD.42.3500. [11] R. Jackiw and S.-Y. Pi, Soliton solutions to the gauged nonlinear Schrödinger equation on the plane, Phys. Rev. Lett., 64 (1990), 2969-2972.  doi: 10.1103/PhysRevLett.64.2969. [12] M.-J. Kang and A. Vasseur, Asymptotic analysis of Vlasov-type equations under strong local alignment regime, Math. Models Methods Appl. Sci., 25 (2015), 2153-2173.  doi: 10.1142/S0218202515500542. [13] C. Landim, Hydrodynamic limit of interacting particle systems,, School and Conference on Probability Theory, ICTP Lect. Notes, XVII, Abdus Salam Int. Cent. Theoret. Phys., Trieste, (2004), 57–100. [14] C. C. Lee and T. C. Lin, Incompressible and compressible limits of two-component Gross-Pitaevskii equations with rotating fields and trap potentials, J. Math. Phys., 49 (2008), 043517.  doi: 10.1063/1.2912716. [15] H. L. Li and C. K. Lin, Zero Debye length asymptotic of the quantum hydrodynamic model of semiconductors, Comm. Math. Phys., 256 (2005), 195-212.  doi: 10.1007/s00220-005-1316-7. [16] Z. M. Lim, Large data well-posedness in the energy space of the Chern-Simons-Schrödinger system, J. Differential Equations, 264 (2018), 2553-2597.  doi: 10.1016/j.jde.2017.10.026. [17] P.-L. Lions, Mathematical Topics in Fluid Mechanics, The Clarendon Press, Oxford University Press, New York, 1998. [18] P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl., 77 (1998), 585-627.  doi: 10.1016/S0021-7824(98)80139-6. [19] B. Liu and P. Smith, Global wellposedness of the equivariant Chern-Simons-Schrödinger equation, Rev. Mat. Iberoam., 32 (2016), 751-794.  doi: 10.4171/RMI/898. [20] B. Liu, P. Smith and D. Tataru, Local wellposedness of Chern-Simons-Schrödinger equation, Int. Math. Res. Not. IMRN, 2014 (2014), 6341-6398.  doi: 10.1093/imrn/rnt161. [21] C.-K. Lin and K.-C. Wu, Hydrodynamic limits of the nonlinear Klein-Gordon equation, J. Math. Pures Appl., 98 (2012), 328-345.  doi: 10.1016/j.matpur.2012.02.002. [22] T. C. Lin and P. Zhang, Incompressible and compressible limits of coupled systems of nonlinear Schrödinger equations, Commun. Math. Phys., 266 (2006), 547-569.  doi: 10.1007/s00220-006-0061-x. [23] E. Madelung, Quantentheorie in hydrodynamischer form, Z. Physik, 40 (1927), 322. [24] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7. [25] A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge Univ. Press. 2002. [26] A. Mellet and A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations, Commun. Math. Phys., 281 (2008), 573-596.  doi: 10.1007/s00220-008-0523-4. [27] M. Puel, Convergence of the Schrödinger-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 27 (2002), 2311-2331.  doi: 10.1081/PDE-120016159. [28] L. Saint-Raymond, Convergence of solutions to the Boltzmann equation in the incompressible Euler limit, Arch. Rational Mech. Anal., 166 (2003), 47-80.  doi: 10.1007/s00205-002-0228-3. [29] L. Saint-Raymond, Hydrodynamic limits: Some improvements of the relative entropy methods, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 705-744.  doi: 10.1016/j.anihpc.2008.01.001. [30] H. T. Yau, Relative entropy and hydrodynamics of Ginzberg-Landau models, Lett. Math. Phys., 22 (1991), 63-80.  doi: 10.1007/BF00400379.
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