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

Molinet, Luc; Vento, Stéphane

doi:10.1090/S0002-9947-2012-05505-X

Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: The periodic case
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Trans. Amer. Math. Soc. 365 (2013), 123-141 Request permission

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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