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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: The periodic case


Authors: Luc Molinet and Stéphane Vento
Journal: Trans. Amer. Math. Soc. 365 (2013), 123-141
MSC (2010): Primary 35E15; Secondary 35M11, 35Q53, 35Q60
Published electronically: June 27, 2012
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Abstract: We prove that the KdV-Burgers equation is globally well-posed in $ H^{-1}(\mathbb{T}) $ with a solution-map that is analytic from $ H^{-1}(\mathbb{T}) $ to $ C([0,T];H^{-1}(\mathbb{T}))$, whereas it is ill-posed in $ H^s(\mathbb{T}) $, as soon as $ s<-1 $, in the sense that the flow-map $ u_0\mapsto u(t) $ cannot be continuous from $ H^s(\mathbb{T}) $ to even $ {\mathcal D}'(\mathbb{T}) $ at any fixed $ t>0 $ small enough. In view of the result of Kappeler and Topalov for KdV it thus appears that even if the dissipation part of the KdV-Burgers equation allows us to lower the $ C^\infty $ critical index with respect to the KdV equation, it does not permit us to improve the $ C^0$ critical index.


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Additional Information

Luc Molinet
Affiliation: Laboratoire de Mathématiques et Physique Théorique, Université François Rabelais Tours, Fédération Denis Poisson-CNRS, Parc Grandmont, 37200 Tours, France
Email: Luc.Molinet@lmpt.univ-tours.fr

Stéphane Vento
Affiliation: L.A.G.A., Institut Galilée, Université Paris 13, 93430 Villetaneuse, France
Email: vento@math.univ-paris13.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05505-X
PII: S 0002-9947(2012)05505-X
Received by editor(s): March 28, 2010
Received by editor(s) in revised form: November 15, 2010
Published electronically: June 27, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.