Well-posedness and inviscid limit behavior of solution for the generalized 1D Ginzburg-Landau equation

Zhaohui Huo; Yueling Jia

doi:10.1016/j.matpur.2009.04.003

Well-posedness and inviscid limit behavior of solution for the generalized 1D Ginzburg-Landau equation

Zhaohui Huo
, Yueling Jia

Department of Mathematics

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

7 Citations (Scopus)

Abstract

The Cauchy problem of the one-dimensional generalized Ginzburg-Landau (GGL) equation is considered. The local well-posedness is obtained for initial data in H^s (R) with s > 0, and global result in H^s (R) with s > 0 is also obtained under some conditions. Moreover, the relation between the solution for GGL equation and the solution for the derivative nonlinear Schrödinger (DNLS) equation is studied. It is proved that for some T > 0, the solution of Cauchy problem for the GGL equation converge to the solution of Cauchy problem for the DNLS in the natural space C ([0, T] ; H^s) with s > frac(1, 2) if some coefficients tend to zero. Moreover, if initial data belong to H², the convergence holds in C ([0, T] ; H¹) for any T > 0. © 2009 Elsevier Masson SAS. All rights reserved.

Original language	English
Pages (from-to)	18-51
Journal	Journal des Mathematiques Pures et Appliquees
Volume	92
Issue number	1
DOIs	https://doi.org/10.1016/j.matpur.2009.04.003
Publication status	Published - Jul 2009

Bibliographical note

Publication details (e.g. title, author(s), publication statuses and dates) are captured on an “AS IS” and “AS AVAILABLE” basis at the time of record harvesting from the data source. Suggestions for further amendments or supplementary information can be sent to [email protected].

Funding

The authors want to express their great thanks to the referee for his/her many helpful suggestions and comments. Z. Huo was supported by the NSF of China (No. 10601006). Y. Jia was supported by the NSF of China (No. 10701013). Z. Huo was partially supported by Department of Mathematics of City University of Hong Kong and would like to thank Professor Tong Yang for his invitation and support.

Research Keywords

Derivative nonlinear Schrödinger equation
Generalized 1D Ginzburg-Landau equation
Inviscid limit behavior
Well-posedness

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title = "Well-posedness and inviscid limit behavior of solution for the generalized 1D Ginzburg-Landau equation",

abstract = "The Cauchy problem of the one-dimensional generalized Ginzburg-Landau (GGL) equation is considered. The local well-posedness is obtained for initial data in Hs (R) with s > 0, and global result in Hs (R) with s > 0 is also obtained under some conditions. Moreover, the relation between the solution for GGL equation and the solution for the derivative nonlinear Schr{\"o}dinger (DNLS) equation is studied. It is proved that for some T > 0, the solution of Cauchy problem for the GGL equation converge to the solution of Cauchy problem for the DNLS in the natural space C ([0, T] ; Hs) with s > frac(1, 2) if some coefficients tend to zero. Moreover, if initial data belong to H2, the convergence holds in C ([0, T] ; H1) for any T > 0. {\textcopyright} 2009 Elsevier Masson SAS. All rights reserved.",

keywords = "Derivative nonlinear Schr{\"o}dinger equation, Generalized 1D Ginzburg-Landau equation, Inviscid limit behavior, Well-posedness",

author = "Zhaohui Huo and Yueling Jia",

note = "Publication details (e.g. title, author(s), publication statuses and dates) are captured on an “AS IS” and “AS AVAILABLE” basis at the time of record harvesting from the data source. Suggestions for further amendments or supplementary information can be sent to [email protected].",

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AU - Huo, Zhaohui

AU - Jia, Yueling

N1 - Publication details (e.g. title, author(s), publication statuses and dates) are captured on an “AS IS” and “AS AVAILABLE” basis at the time of record harvesting from the data source. Suggestions for further amendments or supplementary information can be sent to [email protected].

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N2 - The Cauchy problem of the one-dimensional generalized Ginzburg-Landau (GGL) equation is considered. The local well-posedness is obtained for initial data in Hs (R) with s > 0, and global result in Hs (R) with s > 0 is also obtained under some conditions. Moreover, the relation between the solution for GGL equation and the solution for the derivative nonlinear Schrödinger (DNLS) equation is studied. It is proved that for some T > 0, the solution of Cauchy problem for the GGL equation converge to the solution of Cauchy problem for the DNLS in the natural space C ([0, T] ; Hs) with s > frac(1, 2) if some coefficients tend to zero. Moreover, if initial data belong to H2, the convergence holds in C ([0, T] ; H1) for any T > 0. © 2009 Elsevier Masson SAS. All rights reserved.

AB - The Cauchy problem of the one-dimensional generalized Ginzburg-Landau (GGL) equation is considered. The local well-posedness is obtained for initial data in Hs (R) with s > 0, and global result in Hs (R) with s > 0 is also obtained under some conditions. Moreover, the relation between the solution for GGL equation and the solution for the derivative nonlinear Schrödinger (DNLS) equation is studied. It is proved that for some T > 0, the solution of Cauchy problem for the GGL equation converge to the solution of Cauchy problem for the DNLS in the natural space C ([0, T] ; Hs) with s > frac(1, 2) if some coefficients tend to zero. Moreover, if initial data belong to H2, the convergence holds in C ([0, T] ; H1) for any T > 0. © 2009 Elsevier Masson SAS. All rights reserved.

KW - Derivative nonlinear Schrödinger equation

KW - Generalized 1D Ginzburg-Landau equation

KW - Inviscid limit behavior

KW - Well-posedness

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DO - 10.1016/j.matpur.2009.04.003

M3 - RGC 21 - Publication in refereed journal

SN - 0021-7824

VL - 92

SP - 18

EP - 51

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

IS - 1

ER -

Well-posedness and inviscid limit behavior of solution for the generalized 1D Ginzburg-Landau equation

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