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

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.