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