Blow-up profile for the complex Ginzburg-Landau equation
Nader Masmoudi, Hatem Zaag
We construct a solution to the complex Ginzburg-Landau equation, which blows up in finite time T only at one blow-up point. We also give a sharp description of its blow-up profile. The proof relies on the reduction of the problem to a finite-dimensional one, and the use of index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint and it has a second neutral mode. In the last section, the interpretation of the parameters of the finite-dimensional problem in terms of the blow-up time and the blow-up point gives the stability of the constructed solution with respect to perturbations in the initial data.
Journal of Functional Analysis
https://doi.org/10.1016/j.jfa.2008.03.008
Blow-up profile
Blow-up solution
Complex Ginzburg-Landau equation
10.1016/j.jfa.2008.03.008
Dive into the research topics of 'Blow-up profile for the complex Ginzburg-Landau equation'. Together they form a unique fingerprint.
Complex Ginzburg-Landau Equation Mathematics 100%
Blow-up Mathematics 64%
Profile Mathematics 62%
Index Theory Mathematics 22%
Blow-up Time Mathematics 22%
Interpretation Mathematics 12%
Perturbation Mathematics 12%
Operator Mathematics 8%
Masmoudi, N., & Zaag, H. (2008). Blow-up profile for the complex Ginzburg-Landau equation. Journal of Functional Analysis, 255(7), 1613-1666. https://doi.org/10.1016/j.jfa.2008.03.008
Blow-up profile for the complex Ginzburg-Landau equation. / Masmoudi, Nader; Zaag, Hatem.
In: Journal of Functional Analysis, Vol. 255, No. 7, 01.10.2008, p. 1613-1666.
Masmoudi, N & Zaag, H 2008, 'Blow-up profile for the complex Ginzburg-Landau equation', Journal of Functional Analysis, vol. 255, no. 7, pp. 1613-1666. https://doi.org/10.1016/j.jfa.2008.03.008
Masmoudi N, Zaag H. Blow-up profile for the complex Ginzburg-Landau equation. Journal of Functional Analysis. 2008 Oct 1;255(7):1613-1666. https://doi.org/10.1016/j.jfa.2008.03.008
Masmoudi, Nader ; Zaag, Hatem. / Blow-up profile for the complex Ginzburg-Landau equation. In: Journal of Functional Analysis. 2008 ; Vol. 255, No. 7. pp. 1613-1666.
@article{acf44ce40aba4a199489ac42ef710f35,
title = "Blow-up profile for the complex Ginzburg-Landau equation",
abstract = "We construct a solution to the complex Ginzburg-Landau equation, which blows up in finite time T only at one blow-up point. We also give a sharp description of its blow-up profile. The proof relies on the reduction of the problem to a finite-dimensional one, and the use of index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint and it has a second neutral mode. In the last section, the interpretation of the parameters of the finite-dimensional problem in terms of the blow-up time and the blow-up point gives the stability of the constructed solution with respect to perturbations in the initial data.",
keywords = "Blow-up profile, Blow-up solution, Complex Ginzburg-Landau equation, Stability",
author = "Nader Masmoudi and Hatem Zaag",
note = "Funding Information: * Corresponding author. E-mail addresses: masmoudi@cims.nyu.edu (N. Masmoudi), zaag@math.univ-paris13.fr (H. Zaag). 1 This author was partially supported by an NSF grant DMS-0703145. 2 This author received support from the French Agence Nationale de la Recherche, project ONDENONLIN, reference ANR-06-BLAN-0185.",
doi = "10.1016/j.jfa.2008.03.008",
journal = "Journal of Functional Analysis",
T1 - Blow-up profile for the complex Ginzburg-Landau equation
AU - Masmoudi, Nader
AU - Zaag, Hatem
N1 - Funding Information: * Corresponding author. E-mail addresses: masmoudi@cims.nyu.edu (N. Masmoudi), zaag@math.univ-paris13.fr (H. Zaag). 1 This author was partially supported by an NSF grant DMS-0703145. 2 This author received support from the French Agence Nationale de la Recherche, project ONDENONLIN, reference ANR-06-BLAN-0185.
N2 - We construct a solution to the complex Ginzburg-Landau equation, which blows up in finite time T only at one blow-up point. We also give a sharp description of its blow-up profile. The proof relies on the reduction of the problem to a finite-dimensional one, and the use of index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint and it has a second neutral mode. In the last section, the interpretation of the parameters of the finite-dimensional problem in terms of the blow-up time and the blow-up point gives the stability of the constructed solution with respect to perturbations in the initial data.
AB - We construct a solution to the complex Ginzburg-Landau equation, which blows up in finite time T only at one blow-up point. We also give a sharp description of its blow-up profile. The proof relies on the reduction of the problem to a finite-dimensional one, and the use of index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint and it has a second neutral mode. In the last section, the interpretation of the parameters of the finite-dimensional problem in terms of the blow-up time and the blow-up point gives the stability of the constructed solution with respect to perturbations in the initial data.
KW - Blow-up profile
KW - Blow-up solution
KW - Complex Ginzburg-Landau equation
U2 - 10.1016/j.jfa.2008.03.008
DO - 10.1016/j.jfa.2008.03.008
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis