Sharp ill-posedness results for the KdV and mKdV equations on the torus

We establish a new a priori bound for L2L2-bounded sequences of solutions to the mKdV equations on the torus. This first enables us to construct weak solutions in L2L2 for this equation and to check that the “solutions” constructed by Kappeler and Topalov in the defocusing case satisfy the equation in some weak sense. In a second time, we prove that the solution-maps associated with the mKdV and the KdV equations are discontinuous for the Hs(T)Hs(T) topology for respectively, s<0s<0 and s<−1s<−1. These last results are sharp.