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

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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

Abstract
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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).
[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.
[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.
[4]	R. Hempel and J. Voigt, On the $L_p$-spectrum of Schrödinger operators,, J. Math. Anal. Appl. 121 (1987), 121 (1987), 138.
[5]	C. Huang and B. Wang, Inviscid limit for the energy-critical complex Ginzburg-Landau equation,, J. Funct. Anal. 255 (2008), 255 (2008), 681.
[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).
[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.
[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.
[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.
[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.
[11]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, 44 (1983).

show all references

References:

[1]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext. Springer, (2011).
[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.
[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.
[4]	R. Hempel and J. Voigt, On the $L_p$-spectrum of Schrödinger operators,, J. Math. Anal. Appl. 121 (1987), 121 (1987), 138.
[5]	C. Huang and B. Wang, Inviscid limit for the energy-critical complex Ginzburg-Landau equation,, J. Funct. Anal. 255 (2008), 255 (2008), 681.
[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).
[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.
[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.
[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.
[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.
[11]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, 44 (1983).

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