On the regularization mechanism for the periodic Korteweg-de Vries equation

In this paper we develop and use successive averaging methods for explaining the regularization mechanism in the the periodic Korteweg–de Vries (KdV) equation in the homogeneous Sobolev spaces Ḣs for s ≥ 0. Specifically, we prove the global existence, uniqueness, and Lipschitz‐continuous dependence...

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Bibliographic Details
Published in:Communications on pure and applied mathematics 2011-05, Vol.64 (5), p.591-648
Main Authors: Babin, Anatoli V., Ilyin, Alexei A., Titi, Edriss S.
Format: Article
Language:English
Subjects:
Exact sciences and technology
General mathematics
General, history and biography
Mathematical analysis
Mathematical models
Mathematics
Numerical analysis
Numerical analysis in abstract spaces
Numerical analysis. Scientific computation
Numerical linear algebra
Ordinary differential equations
Sciences and techniques of general use
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Summary:In this paper we develop and use successive averaging methods for explaining the regularization mechanism in the the periodic Korteweg–de Vries (KdV) equation in the homogeneous Sobolev spaces Ḣs for s ≥ 0. Specifically, we prove the global existence, uniqueness, and Lipschitz‐continuous dependence on the initial data of the solutions of the periodic KdV. For the case where the initial data is in L2 we also show the Lipschitz‐continuous dependence of these solutions with respect to the initial data as maps from Ḣs to Ḣs for s ∈(−1,0]. © 2010 Wiley Periodicals, Inc.
ISSN:0010-3640
1097-0312
DOI:10.1002/cpa.20356