# American Institute of Mathematical Sciences

June  2012, 32(6): 2233-2251. doi: 10.3934/dcds.2012.32.2233

## On the fluid dynamical approximation to the nonlinear Klein-Gordon equation

 1 Department of Applied Mathematics and, Center of Mathematical Modeling and Scientiﬁc Computing, National Chiao Tung University, Hsinchu 30010, Taiwan 2 Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB3 OWA, United Kingdom

Received  February 2011 Revised  December 2011 Published  February 2012

We study the nonrelativistic, semiclassical and nonrelativistic-semiclassical limits of the (modulated) nonlinear Klein-Gordon equations from its hydrodynamical structure via WKB analysis. The nonrelativistic-semiclassical limit is proved rigorously by modulated energy method.
Citation: Chi-Kun Lin, Kung-Chien Wu. On the fluid dynamical approximation to the nonlinear Klein-Gordon equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2233-2251. doi: 10.3934/dcds.2012.32.2233
##### References:
 [1] Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations,, Commun. in Partial Differential Equations, 25 (2000), 737. doi: 10.1080/03605300008821529. [2] R. Carles, Geometric optics and instability for semi-classical Schrödinger equations,, Arch. Rational Mech. Anal., 183 (2007), 525. doi: 10.1007/s00205-006-0017-5. [3] Q. Chang, Y.-S. Wong and C.-K. Lin, Numerical computation for long-wave short-wave interaction equations in semi-classical limit,, Journal of Computational Physics, 227 (2008), 8489. doi: 10.1016/j.jcp.2008.05.015. [4] T. Colin and A. Soyeur, Some singular limits for evolutionary Ginzburg-Landau equations,, Asymptotic Analysis, 13 (1996), 361. [5] B. Desjardins, C.-K. Lin and T.-C. Tso, Semiclassical limit of the derivative nonlinear Schrödinger equation,, Math. Models Methods Appl. Sci., 10 (2000), 261. [6] B. Desjardins and C.-K. Lin, On the semiclassical limit of the general modified NLS equation,, J. Math. Anal. Appl., 260 (2001), 546. [7] N. Ercolani and R. Montgometry, On the fluid approximation to a nonlinear Schrödinger equation,, Physics Letters A, 180 (1993), 402. doi: 10.1016/0375-9601(93)90290-G. [8] I. Gasser, C.-K. Lin and P. Markowich, A review of dispersive limit of the (non)linear Schrödinger-type equation,, Taiwanese J. of Mathematics., 4 (2000), 501. [9] V. L. Ginzburg and L. P. Pitaevskiĭ, On the theory of superfluidity,, Sov. Phys. JETP, 34(7) (1958), 858. [10] E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time,, Proc. Amer. Math. Soc., 126 (1998), 523. doi: 10.1090/S0002-9939-98-04164-1. [11] S. Jin, C. D. Levermore and D. W. McLaughlin, The semiclassical limit of the defocusing NLS hierarchy,, Comm. Pure Appl. Math., 52 (1999), 613. doi: 10.1002/(SICI)1097-0312(199905)52:5<613::AID-CPA2>3.0.CO;2-L. [12] J.-H. Lee and C.-K. Lin, The behavior of solutions of NLS equation of derivative type in the semiclassical limit,, Chaos, 13 (2002), 1475. doi: 10.1016/S0960-0779(01)00157-6. [13] J.-H. Lee, C.-K. Lin and O. K. Pashaev, Shock waves, chiral solitons and semiclassical limit of one-dimensional anyons,, Chaos, 19 (2004), 109. doi: 10.1016/S0960-0779(03)00084-5. [14] H.-L. Li and C.-K. Lin, Semiclassical limit and well-posedness of nonlinear Schrödinger-Poisson systems,, Electronic Journal of Differential Equations, 2003 (). [15] H.-L. Li and C.-K. Lin, Zero Debye length asymptotic of the quantum hydrodynamic model of semiconductors,, Commun. Math. Phys., 256 (2005), 195. doi: 10.1007/s00220-005-1316-7. [16] C.-K. Lin and Y.-S. Wong, Zero-dispersion limit of the short-wave-long-wave interaction equations,, Journal of Differential Equations, 228 (2006), 87. doi: 10.1016/j.jde.2006.03.027. [17] C.-K. Lin and K.-C. Wu, Singular limits of the Klein-Gordon equation,, Arch. Rational Mech. Anal., 197 (2010), 689. doi: 10.1007/s00205-010-0324-8. [18] F.-H. Lin and J. X. Xin, On the incompressible fluid limit and the vortex motion law of the nonlinear Schrödinger equation,, Commun. in Math. Phys., 200 (1999), 249. doi: 10.1007/s002200050529. [19] T.-C. Lin and P. Zhang, Incompressible and compressible limit of coupled systems of nonlinear Schrödinger equations,, Commun. Math. Phys., 266 (2006), 547. [20] S. Machihara, K. Nakanishi and T. Ozawa, Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations,, Mathematische Annalen, 322 (2002), 603. doi: 10.1007/s002080200008. [21] A. Messiah, "Quantum Mechanics,", Vol. 1 & 2, (1999). [22] H. M. Pilkuhn, "Relativistic Quantum Mechanics,", Texts and Monographs in Physics, (2003). [23] M. Puel, Convergence of the Schrödinger-Poisson system to the incompressible Euler equations,, Commun. in Partial Differential Equations, 27 (2002), 2311. doi: 10.1081/PDE-120016159. [24] J. Shatah and M. Struwe, "Geometric Wave Equations,", Courant Lecture Notes in Mathematics, (1998). [25] C. Sulem and P.-L. Sulem, "The Nonlinear Schrödinger Equation. Self-focusing and Wave Collapse,", Appl. Math. Sci., 139 (1999). [26] W. Strauss, "Nonlinear Wave Equations,", CBMS Regional Conference Series in Mathematics, 73 (1989). [27] K.-C. Wu, Convergence of the Klein-Gordon equation to the wave map equation with magnetic field,, J. Math. Anal. Appl., 365 (2010), 584. doi: 10.1016/j.jmaa.2009.11.022.

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##### References:
 [1] Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations,, Commun. in Partial Differential Equations, 25 (2000), 737. doi: 10.1080/03605300008821529. [2] R. Carles, Geometric optics and instability for semi-classical Schrödinger equations,, Arch. Rational Mech. Anal., 183 (2007), 525. doi: 10.1007/s00205-006-0017-5. [3] Q. Chang, Y.-S. Wong and C.-K. Lin, Numerical computation for long-wave short-wave interaction equations in semi-classical limit,, Journal of Computational Physics, 227 (2008), 8489. doi: 10.1016/j.jcp.2008.05.015. [4] T. Colin and A. Soyeur, Some singular limits for evolutionary Ginzburg-Landau equations,, Asymptotic Analysis, 13 (1996), 361. [5] B. Desjardins, C.-K. Lin and T.-C. Tso, Semiclassical limit of the derivative nonlinear Schrödinger equation,, Math. Models Methods Appl. Sci., 10 (2000), 261. [6] B. Desjardins and C.-K. Lin, On the semiclassical limit of the general modified NLS equation,, J. Math. Anal. Appl., 260 (2001), 546. [7] N. Ercolani and R. Montgometry, On the fluid approximation to a nonlinear Schrödinger equation,, Physics Letters A, 180 (1993), 402. doi: 10.1016/0375-9601(93)90290-G. [8] I. Gasser, C.-K. Lin and P. Markowich, A review of dispersive limit of the (non)linear Schrödinger-type equation,, Taiwanese J. of Mathematics., 4 (2000), 501. [9] V. L. Ginzburg and L. P. Pitaevskiĭ, On the theory of superfluidity,, Sov. Phys. JETP, 34(7) (1958), 858. [10] E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time,, Proc. Amer. Math. Soc., 126 (1998), 523. doi: 10.1090/S0002-9939-98-04164-1. [11] S. Jin, C. D. Levermore and D. W. McLaughlin, The semiclassical limit of the defocusing NLS hierarchy,, Comm. Pure Appl. Math., 52 (1999), 613. doi: 10.1002/(SICI)1097-0312(199905)52:5<613::AID-CPA2>3.0.CO;2-L. [12] J.-H. Lee and C.-K. Lin, The behavior of solutions of NLS equation of derivative type in the semiclassical limit,, Chaos, 13 (2002), 1475. doi: 10.1016/S0960-0779(01)00157-6. [13] J.-H. Lee, C.-K. Lin and O. K. Pashaev, Shock waves, chiral solitons and semiclassical limit of one-dimensional anyons,, Chaos, 19 (2004), 109. doi: 10.1016/S0960-0779(03)00084-5. [14] H.-L. Li and C.-K. Lin, Semiclassical limit and well-posedness of nonlinear Schrödinger-Poisson systems,, Electronic Journal of Differential Equations, 2003 (). [15] H.-L. Li and C.-K. Lin, Zero Debye length asymptotic of the quantum hydrodynamic model of semiconductors,, Commun. Math. Phys., 256 (2005), 195. doi: 10.1007/s00220-005-1316-7. [16] C.-K. Lin and Y.-S. Wong, Zero-dispersion limit of the short-wave-long-wave interaction equations,, Journal of Differential Equations, 228 (2006), 87. doi: 10.1016/j.jde.2006.03.027. [17] C.-K. Lin and K.-C. Wu, Singular limits of the Klein-Gordon equation,, Arch. Rational Mech. Anal., 197 (2010), 689. doi: 10.1007/s00205-010-0324-8. [18] F.-H. Lin and J. X. Xin, On the incompressible fluid limit and the vortex motion law of the nonlinear Schrödinger equation,, Commun. in Math. Phys., 200 (1999), 249. doi: 10.1007/s002200050529. [19] T.-C. Lin and P. Zhang, Incompressible and compressible limit of coupled systems of nonlinear Schrödinger equations,, Commun. Math. Phys., 266 (2006), 547. [20] S. Machihara, K. Nakanishi and T. Ozawa, Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations,, Mathematische Annalen, 322 (2002), 603. doi: 10.1007/s002080200008. [21] A. Messiah, "Quantum Mechanics,", Vol. 1 & 2, (1999). [22] H. M. Pilkuhn, "Relativistic Quantum Mechanics,", Texts and Monographs in Physics, (2003). [23] M. Puel, Convergence of the Schrödinger-Poisson system to the incompressible Euler equations,, Commun. in Partial Differential Equations, 27 (2002), 2311. doi: 10.1081/PDE-120016159. [24] J. Shatah and M. Struwe, "Geometric Wave Equations,", Courant Lecture Notes in Mathematics, (1998). [25] C. Sulem and P.-L. Sulem, "The Nonlinear Schrödinger Equation. Self-focusing and Wave Collapse,", Appl. Math. Sci., 139 (1999). [26] W. Strauss, "Nonlinear Wave Equations,", CBMS Regional Conference Series in Mathematics, 73 (1989). [27] K.-C. Wu, Convergence of the Klein-Gordon equation to the wave map equation with magnetic field,, J. Math. Anal. Appl., 365 (2010), 584. doi: 10.1016/j.jmaa.2009.11.022.
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