American Institute of Mathematical Sciences

Advanced
2015, 2015(special): 754-763. doi: 10.3934/proc.2015.0754

Solvability of a class of complex Ginzburg-Landau equations in periodic Sobolev spaces

Yuta Kugo 1, , Motohiro Sobajima 2, , Toshiyuki Suzuki 1, , Tomomi Yokota 1, and  Kentarou Yoshii 1,

1. 

Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan, Japan, Japan
2. 

Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo

Received  August 2014 Revised  January 2015 Published  November 2015

This paper is concerned with the Cauchy problem for the complex Ginzburg-Landau type equation $u_t = (\delta _{1}+i\delta _{2})\Delta u -i\mu |u| ^{2\sigma}u$ in $(0,\infty)\times\mathbb{R}^d$, where $\delta_{1}>0$, $\delta_{2}, \mu \in \mathbb{R}$ and $d\in\mathbb{N}$. Existence and uniqueness of spatially periodic solutions to the problem are established in a space which corresponds to the Sobolev space on the $d$-dimensional torus when $0<\sigma<\infty$ ($d=1, 2$) and $0<\sigma<1/(d-2)$ ($d \ge 3$). The result improves the case $p=2$ of the result in the space $W^{1,p}$ given by Gao-Wang [2，Theorem 1] in which it is assumed that $d < p$ and $\sigma < p/d$.
Keywords: spatially periodic solutions, Complex Ginzburg-Landau equations, global existence..
Mathematics Subject Classification: Primary: 35K55; Secondary: 35Q5.
Citation: Yuta Kugo, Motohiro Sobajima, Toshiyuki Suzuki, Tomomi Yokota, Kentarou Yoshii. Solvability of a class of complex Ginzburg-Landau equations in periodic Sobolev spaces. Conference Publications, 2015, 2015 (special) : 754-763. doi: 10.3934/proc.2015.0754
References:
[1]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext. Springer, (2011).   Google Scholar

[2]

H. Gao and X. Wang, On the global existence and small dispersion limit for a class of complex Ginzburg-Landau equations,, Math. Methods Appl. Sci. 32 (2009), 32 (2009), 1396.   Google Scholar

[3]

J. Ginibre and G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. I. Compactness methods,, Phys. D 95 (1996), 95 (1996), 191.   Google Scholar

[4]

R. Hempel and J. Voigt, On the $L_p$-spectrum of Schrödinger operators,, J. Math. Anal. Appl. 121 (1987), 121 (1987), 138.   Google Scholar

[5]

C. Huang and B. Wang, Inviscid limit for the energy-critical complex Ginzburg-Landau equation,, J. Funct. Anal. 255 (2008), 255 (2008), 681.   Google Scholar

[6]

V. A Kozlov, V. G. Maz'ya and J. Rossmann, Elliptic boundary value problems in domains with point singularities,, Mathematical Surveys and Monographs 52, 52 (1997).   Google Scholar

[7]

C. D. Levermore and M. Oliver, The complex Ginzburg-Landau equation as a model problem,, Dynamical systems and probabilistic methods in partial differential equations (Berkeley, (1994), 141.   Google Scholar

[8]

T. Ogawa and T. Yokota, Uniqueness and inviscid limits of solutions for the complex Ginzburg-Landau equation in a two-dimensional domain,, Comm. Math. Phys. 245 (2004), 245 (2004), 105.   Google Scholar

[9]

N. Okazawa and T. Yokota, Global existence and smoothing effect for the complex Ginzburg-Landau equation with $p$-Laplacian,, J. Differential Equations 182 (2002), 182 (2002), 541.   Google Scholar

[10]

N. Okazawa and T. Yokota, Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation,, Discrete Contin. Dyn. Syst. 28 (2010), 28 (2010), 311.   Google Scholar

[11]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, 44 (1983).   Google Scholar

show all references

References:
[1]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext. Springer, (2011).   Google Scholar

[2]

H. Gao and X. Wang, On the global existence and small dispersion limit for a class of complex Ginzburg-Landau equations,, Math. Methods Appl. Sci. 32 (2009), 32 (2009), 1396.   Google Scholar

[3]

J. Ginibre and G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. I. Compactness methods,, Phys. D 95 (1996), 95 (1996), 191.   Google Scholar

[4]

R. Hempel and J. Voigt, On the $L_p$-spectrum of Schrödinger operators,, J. Math. Anal. Appl. 121 (1987), 121 (1987), 138.   Google Scholar

[5]

C. Huang and B. Wang, Inviscid limit for the energy-critical complex Ginzburg-Landau equation,, J. Funct. Anal. 255 (2008), 255 (2008), 681.   Google Scholar

[6]

V. A Kozlov, V. G. Maz'ya and J. Rossmann, Elliptic boundary value problems in domains with point singularities,, Mathematical Surveys and Monographs 52, 52 (1997).   Google Scholar

[7]

C. D. Levermore and M. Oliver, The complex Ginzburg-Landau equation as a model problem,, Dynamical systems and probabilistic methods in partial differential equations (Berkeley, (1994), 141.   Google Scholar

[8]

T. Ogawa and T. Yokota, Uniqueness and inviscid limits of solutions for the complex Ginzburg-Landau equation in a two-dimensional domain,, Comm. Math. Phys. 245 (2004), 245 (2004), 105.   Google Scholar

[9]

N. Okazawa and T. Yokota, Global existence and smoothing effect for the complex Ginzburg-Landau equation with $p$-Laplacian,, J. Differential Equations 182 (2002), 182 (2002), 541.   Google Scholar

[10]

N. Okazawa and T. Yokota, Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation,, Discrete Contin. Dyn. Syst. 28 (2010), 28 (2010), 311.   Google Scholar

[11]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, 44 (1983).   Google Scholar

[1]

Sen-Zhong Huang, Peter Takáč. Global smooth solutions of the complex Ginzburg-Landau equation and their dynamical properties. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 825-848. doi: 10.3934/dcds.1999.5.825
[2]

N. I. Karachalios, H. E. Nistazakis, A. N. Yannacopoulos. Remarks on the asymptotic behavior of solutions of complex discrete Ginzburg-Landau equations. Conference Publications, 2005, 2005 (Special) : 476-486. doi: 10.3934/proc.2005.2005.476
[3]

Hongzi Cong, Jianjun Liu, Xiaoping Yuan. Quasi-periodic solutions for complex Ginzburg-Landau equation of nonlinearity $|u|^{2p}u$. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 579-600. doi: 10.3934/dcdss.2010.3.579
[4]

N. I. Karachalios, Hector E. Nistazakis, Athanasios N. Yannacopoulos. Asymptotic behavior of solutions of complex discrete evolution equations: The discrete Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 711-736. doi: 10.3934/dcds.2007.19.711
[5]

Hans G. Kaper, Peter Takáč. Bifurcating vortex solutions of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 871-880. doi: 10.3934/dcds.1999.5.871
[6]

Noboru Okazawa, Tomomi Yokota. Smoothing effect for generalized complex Ginzburg-Landau equations in unbounded domains. Conference Publications, 2001, 2001 (Special) : 280-288. doi: 10.3934/proc.2001.2001.280
[7]

Shijin Ding, Qiang Du. The global minimizers and vortex solutions to a Ginzburg-Landau model of superconducting films. Communications on Pure & Applied Analysis, 2002, 1 (3) : 327-340. doi: 10.3934/cpaa.2002.1.327
[8]

Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345
[9]

Bixiang Wang, Shouhong Wang. Gevrey class regularity for the solutions of the Ginzburg-Landau equations of superconductivity. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 507-522. doi: 10.3934/dcds.1998.4.507
[10]

Dingshi Li, Xiaohu Wang. Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 449-465. doi: 10.3934/dcdsb.2018181
[11]

Dingshi Li, Lin Shi, Xiaohu Wang. Long term behavior of stochastic discrete complex Ginzburg-Landau equations with time delays in weighted spaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5121-5148. doi: 10.3934/dcdsb.2019046
[12]

Hans G. Kaper, Bixiang Wang, Shouhong Wang. Determining nodes for the Ginzburg-Landau equations of superconductivity. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 205-224. doi: 10.3934/dcds.1998.4.205
[13]

Noboru Okazawa, Tomomi Yokota. Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 311-341. doi: 10.3934/dcds.2010.28.311
[14]

Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 665-682. doi: 10.3934/cpaa.2005.4.665
[15]

Alessia Berti, Valeria Berti, Ivana Bochicchio. Global and exponential attractors for a Ginzburg-Landau model of superfluidity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 247-271. doi: 10.3934/dcdss.2011.4.247
[16]

Dmitry Turaev, Sergey Zelik. Analytical proof of space-time chaos in Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1713-1751. doi: 10.3934/dcds.2010.28.1713
[17]

Gregory A. Chechkin, Vladimir V. Chepyzhov, Leonid S. Pankratov. Homogenization of trajectory attractors of Ginzburg-Landau equations with randomly oscillating terms. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1133-1154. doi: 10.3934/dcdsb.2018145
[18]

Kolade M. Owolabi, Edson Pindza. Numerical simulation of multidimensional nonlinear fractional Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 835-851. doi: 10.3934/dcdss.2020048
[19]

Michael Stich, Carsten Beta. Standing waves in a complex Ginzburg-Landau equation with time-delay feedback. Conference Publications, 2011, 2011 (Special) : 1329-1334. doi: 10.3934/proc.2011.2011.1329
[20]

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

 Impact Factor: 

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences