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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Transactions of the American Mathematical Society, American Mathematical Society, 2013, 365 (1), pp.123-141
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Dernière modification le : jeudi 7 février 2019 - 17:52:34
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  • HAL Id : hal-00467657, version 2
  • ARXIV : 1005.4805

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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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