Inviscid limit behavior of solution for the multi-dimensional derivative complex Ginzburg-Landau equation

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March 2014, 7(1): 57-77. doi: 10.3934/krm.2014.7.57

Inviscid limit behavior of solution for the multi-dimensional derivative complex Ginzburg-Landau equation

Yueling Jia ^1, and Zhaohui Huo ^2,

1.	Institute of Applied Physics and Computational Mathematics, Beijing, 100094, China
2.	Institute of Mathematics, Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing, 100190, China

Received September 2013 Revised October 2013 Published December 2013

Abstract
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The inviscid limit behavior of solution is considered for the multi-dimensional derivative complex Ginzburg-Landau(DCGL) equation. For small initial data, it is proved that for some $T>0$, solution of the DCGL equation converges to the solution of the derivative nonlinear Schrödinger (DNLS) equation in natural space $C([0,T]; H^s)(s\geq \frac{n}{2})$ if some coefficients tend to zero.

Keywords: the DNLS equation, Inviscid limit behavior, the DCGL equation..

Mathematics Subject Classification: Primary: 35E15, 35K15, 35K55; Secondary: 35Q5.

Citation: Yueling Jia, Zhaohui Huo. Inviscid limit behavior of solution for the multi-dimensional derivative complex Ginzburg-Landau equation. Kinetic and Related Models, 2014, 7 (1) : 57-77. doi: 10.3934/krm.2014.7.57

References:

[1]	J.Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part I: Schrödinger equation, part II: The KdV equation, Geom. Funct. Anal., 3 (1993), 107-156, 209-262.
[2]	H. R. Brand and R. J. Deissler, Interaction of localized solutions for subcritical bifurcations, Phys. Rev. Lett., 63 (1989), 2801-2804.
[3]	M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys., 65 (1993), 851-1112. doi: 10.1103/RevModPhys.65.851.
[4]	R. J. Deissler and H. R. Brand, Generation of counterpropagating nonlinear interacting traveling waves by localized noise, Phys. Lett. A, 130 (1988), 293-298. doi: 10.1016/0375-9601(88)90613-5.
[5]	A. Doelman and W. Eckhaus, Periodic and quasi-periodic solutions of degenerate modulation equations, Phys. D, 53 (1991), 249-266. doi: 10.1016/0167-2789(91)90065-H.
[6]	J. Q. Duan and P. Holmes, On the Cauchy problem of a generalized Ginzburg-Landau equation, Nonlinear Anal., 22 (1994), 1033-1040. doi: 10.1016/0362-546X(94)90065-5.
[7]	H. G. Feichtinger, Modulation spaces on locally compact Abelian group, Proc. Internat. Conf. on Wavelet and Applications, (2003), 99-140.
[8]	H. J. Gao and J. Q. Duan, On the initial-value problem for the generalized two-dimensional Ginzburg-Landau equation, J. Math. Anal. Appl., 216 (1997), 536-548. doi: 10.1006/jmaa.1997.5682.
[9]	L. Han, B. Wang and B. Guo, Inviscid limit for the derivative Ginzburg-Landau equation with small data in higher spatial dimensions, Print, arXiv:1004.1221.
[10]	Z. H. Huo and Y. L. Jia, Well-posedness and inviscid limit behavior of solution for the generalized 1D Ginzburg-Landau equation, J. Math. Pures Appl., 92 (2009), 18-51. doi: 10.1016/j.matpur.2009.04.003.
[11]	Z. H. Huo and Y. L. Jia, Global well-posedness for the generalized 2D Ginzburg-Landau equation, J. Differ. Eqns., 247 (2009), 260-276. doi: 10.1016/j.jde.2009.03.015.
[12]	A. D. Ionescu and C. E. Kenig, Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, J. Amer. Math. Soc., 20 (2007), 753-798. doi: 10.1090/S0894-0347-06-00551-0.
[13]	Y. L. Jia, Inviscid limit behavior of generalized 1-D Ginzburg-Landau, submitted.
[14]	C. E. Kenig, G. Ponce, C. Rolvent and L. Vega, The genreal quasilinear untrahyperbolic Schrodinger equation, Advances in Mathematics, 206 (2006), 402-433. doi: 10.1016/j.aim.2005.09.005.
[15]	C. E. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 255-288.
[16]	C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405.
[17]	C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090/S0894-0347-96-00200-7.
[18]	C. E. Kenig, G. Ponce and L. Vega, Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations, Invent. Math., 134 (1998), 489-545. doi: 10.1007/s002220050272.
[19]	C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for quasi-linear Schrodinger equations, Invent. Math., 158 (2004), 343-388. doi: 10.1007/s00222-004-0373-4.
[20]	Y. S. Li and B. L. Guo, Global existence of solutions to the derivative 2D Ginzburg-Landau equation, J. Math. Anal. Appl., 249 (2000), 412-432. doi: 10.1006/jmaa.2000.6880.
[21]	T. Ozawa and J. Zhai, Global existence of small classical solutions to nonlinear Schröinger equationsm, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 303-331. doi: 10.1016/j.anihpc.2006.11.010.
[22]	H. Takaoka, Well-posedness for the one dimensional Schrödinger equation with the derivative nonlinearity, Adv. Diff. Eq., 4 (1999), 561-680.
[23]	H. Takaoka, Global well-posedness for Schrödinger equation with derivative in a nonlinear term and data in low-order Sobolev Spaces, Elec. J. Diff. Eq., 2001, 23 pp. (electronic).
[24]	T. Tao, Multilinear weighted convolution of $ L^2 $ functions, and applications to nonlinear dispersive equation, Amer.J. Math., 123 (2001), 839-908. doi: 10.1353/ajm.2001.0035.
[25]	B. X. Wang, The Cauchy problem for critical and subcritial semilinear parabolic equations in $L^r$ (II). Initial data in critial Sobolev spaces $H^{-s,r}$, Nonlinear Anal., 52 (2003), 851-868. doi: 10.1016/S0362-546X(02)00136-0.
[26]	B. X. Wang, B. L. Guo and L. F. Zhao, The global well-posedness and spatial decay of solutions for the derivative complex Ginzburg-Landau equation in $H^1$, Nonlinear Anal., 57 (2004), 1059-1076. doi: 10.1016/j.na.2004.03.032.
[27]	B. Wang and Y. Wang, The inviscid limit of the derivative complex Ginzburg-Landau equation, J. Math. Pures Appl., 83 (2004), 477-502. doi: 10.1016/j.matpur.2003.11.002.

show all references

References:

[1]	J.Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part I: Schrödinger equation, part II: The KdV equation, Geom. Funct. Anal., 3 (1993), 107-156, 209-262.
[2]	H. R. Brand and R. J. Deissler, Interaction of localized solutions for subcritical bifurcations, Phys. Rev. Lett., 63 (1989), 2801-2804.
[3]	M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys., 65 (1993), 851-1112. doi: 10.1103/RevModPhys.65.851.
[4]	R. J. Deissler and H. R. Brand, Generation of counterpropagating nonlinear interacting traveling waves by localized noise, Phys. Lett. A, 130 (1988), 293-298. doi: 10.1016/0375-9601(88)90613-5.
[5]	A. Doelman and W. Eckhaus, Periodic and quasi-periodic solutions of degenerate modulation equations, Phys. D, 53 (1991), 249-266. doi: 10.1016/0167-2789(91)90065-H.
[6]	J. Q. Duan and P. Holmes, On the Cauchy problem of a generalized Ginzburg-Landau equation, Nonlinear Anal., 22 (1994), 1033-1040. doi: 10.1016/0362-546X(94)90065-5.
[7]	H. G. Feichtinger, Modulation spaces on locally compact Abelian group, Proc. Internat. Conf. on Wavelet and Applications, (2003), 99-140.
[8]	H. J. Gao and J. Q. Duan, On the initial-value problem for the generalized two-dimensional Ginzburg-Landau equation, J. Math. Anal. Appl., 216 (1997), 536-548. doi: 10.1006/jmaa.1997.5682.
[9]	L. Han, B. Wang and B. Guo, Inviscid limit for the derivative Ginzburg-Landau equation with small data in higher spatial dimensions, Print, arXiv:1004.1221.
[10]	Z. H. Huo and Y. L. Jia, Well-posedness and inviscid limit behavior of solution for the generalized 1D Ginzburg-Landau equation, J. Math. Pures Appl., 92 (2009), 18-51. doi: 10.1016/j.matpur.2009.04.003.
[11]	Z. H. Huo and Y. L. Jia, Global well-posedness for the generalized 2D Ginzburg-Landau equation, J. Differ. Eqns., 247 (2009), 260-276. doi: 10.1016/j.jde.2009.03.015.
[12]	A. D. Ionescu and C. E. Kenig, Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, J. Amer. Math. Soc., 20 (2007), 753-798. doi: 10.1090/S0894-0347-06-00551-0.
[13]	Y. L. Jia, Inviscid limit behavior of generalized 1-D Ginzburg-Landau, submitted.
[14]	C. E. Kenig, G. Ponce, C. Rolvent and L. Vega, The genreal quasilinear untrahyperbolic Schrodinger equation, Advances in Mathematics, 206 (2006), 402-433. doi: 10.1016/j.aim.2005.09.005.
[15]	C. E. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 255-288.
[16]	C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405.
[17]	C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090/S0894-0347-96-00200-7.
[18]	C. E. Kenig, G. Ponce and L. Vega, Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations, Invent. Math., 134 (1998), 489-545. doi: 10.1007/s002220050272.
[19]	C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for quasi-linear Schrodinger equations, Invent. Math., 158 (2004), 343-388. doi: 10.1007/s00222-004-0373-4.
[20]	Y. S. Li and B. L. Guo, Global existence of solutions to the derivative 2D Ginzburg-Landau equation, J. Math. Anal. Appl., 249 (2000), 412-432. doi: 10.1006/jmaa.2000.6880.
[21]	T. Ozawa and J. Zhai, Global existence of small classical solutions to nonlinear Schröinger equationsm, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 303-331. doi: 10.1016/j.anihpc.2006.11.010.
[22]	H. Takaoka, Well-posedness for the one dimensional Schrödinger equation with the derivative nonlinearity, Adv. Diff. Eq., 4 (1999), 561-680.
[23]	H. Takaoka, Global well-posedness for Schrödinger equation with derivative in a nonlinear term and data in low-order Sobolev Spaces, Elec. J. Diff. Eq., 2001, 23 pp. (electronic).
[24]	T. Tao, Multilinear weighted convolution of $ L^2 $ functions, and applications to nonlinear dispersive equation, Amer.J. Math., 123 (2001), 839-908. doi: 10.1353/ajm.2001.0035.
[25]	B. X. Wang, The Cauchy problem for critical and subcritial semilinear parabolic equations in $L^r$ (II). Initial data in critial Sobolev spaces $H^{-s,r}$, Nonlinear Anal., 52 (2003), 851-868. doi: 10.1016/S0362-546X(02)00136-0.
[26]	B. X. Wang, B. L. Guo and L. F. Zhao, The global well-posedness and spatial decay of solutions for the derivative complex Ginzburg-Landau equation in $H^1$, Nonlinear Anal., 57 (2004), 1059-1076. doi: 10.1016/j.na.2004.03.032.
[27]	B. Wang and Y. Wang, The inviscid limit of the derivative complex Ginzburg-Landau equation, J. Math. Pures Appl., 83 (2004), 477-502. doi: 10.1016/j.matpur.2003.11.002.

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