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(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 Hs(T), as soon as s < -1, in the sense that the flowmap u0 ↦ u(t) cannot be continuous from Hs(T) to even 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 us to lower the C∞ critical index with respect to the KdV equation, it does not permit us to improve the C0 critical index.