Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-29T16:27:15.224Z Has data issue: false hasContentIssue false
  • English
  • Français

Global existence for the generalised 2D Ginzburg-Landau equation

Published online by Cambridge University Press:  17 February 2009

Hongjun Gao  and
Keng-Huat Kwek
Hongjun Gao
Affiliation:
Department of Mathematics, Nanjing Normal University, Nanjing 210097, P. R. China; e-mail: gaohj@pine.njnu.edu.cn.
Keng-Huat Kwek
Affiliation:
Department of Mathematics, The National University of Singapore, 10 Kent Ridge Crescent, Singapore119260.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Ginzburg-Landau type complex partial differential equations are simplified mathematical models for various pattern formation systems in mechanics, physics and chemistry. Most work so far has concentrated on Ginzburg-Landau type equations with one spatial variable (1D). In this paper, the authors study a complex generalised Ginzburg-Landau equation with two spatial variables (2D) and fifth-order and cubic terms containing derivatives. Based on detail analysis, sufficient conditions for the existence and uniqueness of global solutions are obtained.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 3 , January 2003 , pp. 381 - 392
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Bartuccelli, M., Constantin, P., Doenng, C., Gibbon, J. D. and Gisselfält, M., “On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation”, Phys. D 44 (1990) 421–444.CrossRefGoogle Scholar
[2]Bartuccelli, M., Gibbon, J. D. and Oliver, M., “Length scales in solutions of the complex Ginzburg-Landau equation”, Phys. D 89 (1996) 267–286.CrossRefGoogle Scholar
[3]Bu, C., “On the Cauchy problem for the 1 + 2 complex Ginzburg-Landau equation”, J. Austral. Math. Soc. Ser. B 36 (1994) 313–324.CrossRefGoogle Scholar
[4]Bu, C., Shull, R. and Zhao, K., “A periodic boundary value problem for a generalized 2D Ginzburg-Landau equation”, Hokkaido Math. J. 27 (1998) 197–211.CrossRefGoogle Scholar
[5]Doelman, A., “On the nonlinear evolution of patterns (modulation equations and their solutions)”, Ph. D. Thesis, University of Utrecht, the Netherlands, 1990.Google Scholar
[6]Doering, C., Gibbon, J. D., Holm, D. and Nicolaenko, B., “Low-dimensional behavior in the complex Ginzburg-Landau equation”, Nonlinearity 1 (1988) 279–309.CrossRefGoogle Scholar
[7]Doering, C. R., Gibbon, J. D. and Levermore, C. D., “Weak and strong solutions of the complex Ginzburg-Landau equation”, Phys. D 71 (1994) 285–318.CrossRefGoogle Scholar
[8]Duan, J. and Holmes, P., “On the Cauchy problem of a generalized Ginzburg-Landau equation”, Nonlinear Anal. 22 (1994) 1033–1040.CrossRefGoogle Scholar
[9]Duan, J., Holmes, P. and Titi, E. S., “Global existence theory for a generalized Ginzburg-Landau equation”, Nonlinearity 5 (1992) 1303–1314.CrossRefGoogle Scholar
[10]Duan, J., Holmes, P. and Titi, E. S., “Regularity approximation and asymptotic dynamics for a generalized Ginzburg-Landau equation”, Nonlinearity 6 (1993) 915–933.CrossRefGoogle Scholar
[11]Gao, H., “Exponential attractors for a generalized Ginzburg-Landau equation”, Appl. Math. Mech. 16 (1995) 877–882.Google Scholar
[12]Gao, H. and Guo, B., “Finite dimensional inertial forms for 1D generalized Ginzburg-Landau equation”, Sci. China Ser. A 25 (1995) 1233–1247.Google Scholar
[13]Gao, H. and Guo, B., “Numbers of determining nodes for a generalized Ginzburg-Landau equation”, Progr. Natur. Sci. 5 (1995) 636–638.Google Scholar
[14]Gao, H. J. and Duan, J., “On the initial value problem for the generalized 2D Ginzburg-Landau equation”, J. Math. Anal. Appl. 216 (1997) 536–548.CrossRefGoogle Scholar
[15]Gao, H. J., Duan, J. and Lin, G., “Global existence and global attractor for the generalized 2D Ginzburg-Landau equation”, J. Math. Anal. Appl., 247 (2000) 198–216.CrossRefGoogle Scholar
[16]Ghidaglia, J.-M. and Héron, B., “Dimension of the attractor associated to the Ginzburg-Landau equation”, Phys. D 28 (1987) 282–304.CrossRefGoogle Scholar
[17]Guo, B. and Gao, H., “Finite dimensional behavior of generalized Ginzburg-Landau equation”. Progr. Natur. Sci. 4 (1994) 423–434.Google Scholar
[18]Guo, B. and Wang, B., “Finite dimensional behavior for the derivative Ginzburg-Landau equation in two spatial dimensions”, Phys. D 89 (1995) 83–99.Google Scholar
[19]Hale, J. K., Asymptotic behavior of dissipative systems (Amer. Math. Soc., Providence, RI, 1988).Google Scholar
[20]Kalantarov, V. K. and Ladyzhenskaya, O. A., “The occurrence of collapse for quasilinear equations of parabolic and hyperbolic type”, J. Soviet Math. 10 (1978) 53–70.CrossRefGoogle Scholar
[21]Levermore, C. D. and Oliver, M., The complex Ginzburg-Landau equation as a model problem, dynamical system and probabilistic methods in partial differential equations, Lect. in Appl. Math. 31 (Amer. Math. Soc., Providence, RI, 1996).Google Scholar
[22]Levine, H. A., “Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Pu t = [Au + F (u)]”, Arch. Rat. Mech. Anal. 51 (1973) 371–386.CrossRefGoogle Scholar
[23]Mielke, A., “The complex Ginzburg-Landau equation on large and unbounded domains: sharper bounds and attractor”, Nonlinearity 10 (1997) 199–222.CrossRefGoogle Scholar
[24]Mielke, A., “Bounded for the solutions of the complex Ginzburg-Landau equation in terms of the dispersion parameters”, Phys. D 117 (1998) 106–116.CrossRefGoogle Scholar
[25]Mielke, A. and Schneider, G., “Attractors for modulation equations on unbounded domains—existence and comparison”, Nonlinearity 8 (1995) 743–768.CrossRefGoogle Scholar
[26]Temam, R., Infinite dimensional dynamical systems in mechanics and physics, 2n ed. (Springer, New York, 1997).CrossRefGoogle Scholar