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

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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