Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: The periodic case
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- by Luc Molinet and Stéphane Vento PDF
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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.References
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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
- Received by editor(s): March 28, 2010
- Received by editor(s) in revised form: November 15, 2010
- Published electronically: June 27, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 123-141
- MSC (2010): Primary 35E15; Secondary 35M11, 35Q53, 35Q60
- DOI: https://doi.org/10.1090/S0002-9947-2012-05505-X
- MathSciNet review: 2984054