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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, New York, 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), 1396-1414. Google Scholar

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J. Ginibre and G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. I. Compactness methods, Phys. D 95 (1996), 191-228. Google Scholar

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R. Hempel and J. Voigt, On the $L_p$-spectrum of Schrödinger operators, J. Math. Anal. Appl. 121 (1987), 138-159. Google Scholar

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C. Huang and B. Wang, Inviscid limit for the energy-critical complex Ginzburg-Landau equation, J. Funct. Anal. 255 (2008), 681-725. Google Scholar

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

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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, CA, 1994), 141-190, Lectures in Appl. Math., 31, Amer. Math. Soc., Providence, RI, 1996. Google Scholar

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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), 105-121. 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), 541-576. 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), 311-341. Google Scholar

[11]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. Google Scholar

show all references

References:
[1]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 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), 1396-1414. 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), 191-228. Google Scholar

[4]

R. Hempel and J. Voigt, On the $L_p$-spectrum of Schrödinger operators, J. Math. Anal. Appl. 121 (1987), 138-159. Google Scholar

[5]

C. Huang and B. Wang, Inviscid limit for the energy-critical complex Ginzburg-Landau equation, J. Funct. Anal. 255 (2008), 681-725. 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, American Mathematical Society, Providence, RI, 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, CA, 1994), 141-190, Lectures in Appl. Math., 31, Amer. Math. Soc., Providence, RI, 1996. 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), 105-121. 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), 541-576. 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), 311-341. Google Scholar

[11]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. Google Scholar

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