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

Abstract : We prove that the KdV-Burgers is globally well-posed in $H^{-1}(\T)$ with a solution-map that is analytic from $H^{-1}(\T)$ to $C([0,T];H^{-1}(\T))$ whereas it is ill-posed in $H^s(\T)$, as soon as $s<-1$, in the sense that the flow-map $u_0\mapsto u(t)$ cannot be continuous from $H^s(\T)$ to even ${\cal D}'(\T)$ 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 to lower the $C^\infty$ critical index with respect to the KdV equation, it does not permit to improve the $C^0$ critical index .
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Journal articles

Cited literature [19 references]

https://hal.archives-ouvertes.fr/hal-00467657
Contributor : Luc Molinet <>
Submitted on : Tuesday, March 30, 2010 - 9:17:58 AM
Last modification on : Thursday, April 29, 2021 - 11:53:33 PM
Long-term archiving on: : Thursday, September 23, 2010 - 12:32:20 PM

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### Identifiers

• HAL Id : hal-00467657, version 2
• ARXIV : 1005.4805

### Citation

Luc Molinet, Stéphane Vento. Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the periodic case. Transactions of the American Mathematical Society, American Mathematical Society, 2013, 365 (1), pp.123-141. ⟨hal-00467657v2⟩

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