# Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the real line case

Abstract : We complete the known results on the local Cauchy problem in Sobolev spaces for the KdV-Burgers equation by proving that this equation is well-posed in $H^{-1}(\R)$ with a solution-map that is analytic from $H^{-1}(\R)$ to $C([0,T];H^{-1}(\R))$ whereas it is ill-posed in $H^s(\R)$, as soon as $s<-1$, in the sense that the flow-map $u_0\mapsto u(t)$ cannot be continuous from $H^s(\R)$ to even ${\cal D}'(\R)$ at any fixed $t>0$ small enough. As far as we know, this is the first result of this type for a dispersive-dissipative equation. The framework we develop here should be very useful to prove similar results for other dispersive-dissipative models
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Cited literature [20 references]

https://hal.archives-ouvertes.fr/hal-00436652
Contributor : Luc Molinet <>
Submitted on : Thursday, December 31, 2009 - 10:56:31 AM
Last modification on : Tuesday, May 5, 2020 - 1:03:21 PM
Document(s) archivé(s) le : Thursday, September 23, 2010 - 11:38:55 AM

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### Identifiers

• HAL Id : hal-00436652, version 2
• ARXIV : 0911.5256

### Citation

Luc Molinet, Stéphane Vento. Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the real line case. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Scuola Normale Superiore 2011, 10 (3), pp.531-560. ⟨hal-00436652v2⟩

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