American Institute of Mathematical Sciences

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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

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 & 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.   Google Scholar

[2]

H. R. Brand and R. J. Deissler, Interaction of localized solutions for subcritical bifurcations,, Phys. Rev. Lett., 63 (1989), 2801.   Google Scholar

[3]

M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium,, Rev. Mod. Phys., 65 (1993), 851.  doi: 10.1103/RevModPhys.65.851.  Google Scholar

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R. J. Deissler and H. R. Brand, Generation of counterpropagating nonlinear interacting traveling waves by localized noise,, Phys. Lett. A, 130 (1988), 293.  doi: 10.1016/0375-9601(88)90613-5.  Google Scholar

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A. Doelman and W. Eckhaus, Periodic and quasi-periodic solutions of degenerate modulation equations,, Phys. D, 53 (1991), 249.  doi: 10.1016/0167-2789(91)90065-H.  Google Scholar

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J. Q. Duan and P. Holmes, On the Cauchy problem of a generalized Ginzburg-Landau equation,, Nonlinear Anal., 22 (1994), 1033.  doi: 10.1016/0362-546X(94)90065-5.  Google Scholar

[7]

H. G. Feichtinger, Modulation spaces on locally compact Abelian group,, Proc. Internat. Conf. on Wavelet and Applications, (2003), 99.   Google Scholar

[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.  doi: 10.1006/jmaa.1997.5682.  Google Scholar

[9]

L. Han, B. Wang and B. Guo, Inviscid limit for the derivative Ginzburg-Landau equation with small data in higher spatial dimensions,, Print, ().   Google Scholar

[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.  doi: 10.1016/j.matpur.2009.04.003.  Google Scholar

[11]

Z. H. Huo and Y. L. Jia, Global well-posedness for the generalized 2D Ginzburg-Landau equation,, J. Differ. Eqns., 247 (2009), 260.  doi: 10.1016/j.jde.2009.03.015.  Google Scholar

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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.  doi: 10.1090/S0894-0347-06-00551-0.  Google Scholar

[13]

Y. L. Jia, Inviscid limit behavior of generalized 1-D Ginzburg-Landau,, submitted., ().   Google Scholar

[14]

C. E. Kenig, G. Ponce, C. Rolvent and L. Vega, The genreal quasilinear untrahyperbolic Schrodinger equation,, Advances in Mathematics, 206 (2006), 402.  doi: 10.1016/j.aim.2005.09.005.  Google Scholar

[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.   Google Scholar

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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.  doi: 10.1002/cpa.3160460405.  Google Scholar

[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.  doi: 10.1090/S0894-0347-96-00200-7.  Google Scholar

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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.  doi: 10.1007/s002220050272.  Google Scholar

[19]

C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for quasi-linear Schrodinger equations,, Invent. Math., 158 (2004), 343.  doi: 10.1007/s00222-004-0373-4.  Google Scholar

[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.  doi: 10.1006/jmaa.2000.6880.  Google Scholar

[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.  doi: 10.1016/j.anihpc.2006.11.010.  Google Scholar

[22]

H. Takaoka, Well-posedness for the one dimensional Schrödinger equation with the derivative nonlinearity,, Adv. Diff. Eq., 4 (1999), 561.   Google Scholar

[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 ().   Google Scholar

[24]

T. Tao, Multilinear weighted convolution of $ L^2 $ functions, and applications to nonlinear dispersive equation,, Amer.J. Math., 123 (2001), 839.  doi: 10.1353/ajm.2001.0035.  Google Scholar

[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.  doi: 10.1016/S0362-546X(02)00136-0.  Google Scholar

[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.  doi: 10.1016/j.na.2004.03.032.  Google Scholar

[27]

B. Wang and Y. Wang, The inviscid limit of the derivative complex Ginzburg-Landau equation,, J. Math. Pures Appl., 83 (2004), 477.  doi: 10.1016/j.matpur.2003.11.002.  Google Scholar

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.   Google Scholar

[2]

H. R. Brand and R. J. Deissler, Interaction of localized solutions for subcritical bifurcations,, Phys. Rev. Lett., 63 (1989), 2801.   Google Scholar

[3]

M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium,, Rev. Mod. Phys., 65 (1993), 851.  doi: 10.1103/RevModPhys.65.851.  Google Scholar

[4]

R. J. Deissler and H. R. Brand, Generation of counterpropagating nonlinear interacting traveling waves by localized noise,, Phys. Lett. A, 130 (1988), 293.  doi: 10.1016/0375-9601(88)90613-5.  Google Scholar

[5]

A. Doelman and W. Eckhaus, Periodic and quasi-periodic solutions of degenerate modulation equations,, Phys. D, 53 (1991), 249.  doi: 10.1016/0167-2789(91)90065-H.  Google Scholar

[6]

J. Q. Duan and P. Holmes, On the Cauchy problem of a generalized Ginzburg-Landau equation,, Nonlinear Anal., 22 (1994), 1033.  doi: 10.1016/0362-546X(94)90065-5.  Google Scholar

[7]

H. G. Feichtinger, Modulation spaces on locally compact Abelian group,, Proc. Internat. Conf. on Wavelet and Applications, (2003), 99.   Google Scholar

[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.  doi: 10.1006/jmaa.1997.5682.  Google Scholar

[9]

L. Han, B. Wang and B. Guo, Inviscid limit for the derivative Ginzburg-Landau equation with small data in higher spatial dimensions,, Print, ().   Google Scholar

[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.  doi: 10.1016/j.matpur.2009.04.003.  Google Scholar

[11]

Z. H. Huo and Y. L. Jia, Global well-posedness for the generalized 2D Ginzburg-Landau equation,, J. Differ. Eqns., 247 (2009), 260.  doi: 10.1016/j.jde.2009.03.015.  Google Scholar

[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.  doi: 10.1090/S0894-0347-06-00551-0.  Google Scholar

[13]

Y. L. Jia, Inviscid limit behavior of generalized 1-D Ginzburg-Landau,, submitted., ().   Google Scholar

[14]

C. E. Kenig, G. Ponce, C. Rolvent and L. Vega, The genreal quasilinear untrahyperbolic Schrodinger equation,, Advances in Mathematics, 206 (2006), 402.  doi: 10.1016/j.aim.2005.09.005.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1002/cpa.3160460405.  Google Scholar

[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.  doi: 10.1090/S0894-0347-96-00200-7.  Google Scholar

[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.  doi: 10.1007/s002220050272.  Google Scholar

[19]

C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for quasi-linear Schrodinger equations,, Invent. Math., 158 (2004), 343.  doi: 10.1007/s00222-004-0373-4.  Google Scholar

[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.  doi: 10.1006/jmaa.2000.6880.  Google Scholar

[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.  doi: 10.1016/j.anihpc.2006.11.010.  Google Scholar

[22]

H. Takaoka, Well-posedness for the one dimensional Schrödinger equation with the derivative nonlinearity,, Adv. Diff. Eq., 4 (1999), 561.   Google Scholar

[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 ().   Google Scholar

[24]

T. Tao, Multilinear weighted convolution of $ L^2 $ functions, and applications to nonlinear dispersive equation,, Amer.J. Math., 123 (2001), 839.  doi: 10.1353/ajm.2001.0035.  Google Scholar

[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.  doi: 10.1016/S0362-546X(02)00136-0.  Google Scholar

[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.  doi: 10.1016/j.na.2004.03.032.  Google Scholar

[27]

B. Wang and Y. Wang, The inviscid limit of the derivative complex Ginzburg-Landau equation,, J. Math. Pures Appl., 83 (2004), 477.  doi: 10.1016/j.matpur.2003.11.002.  Google Scholar

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