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

We establish a new a priori bound for $L^2$-bounded sequences of solutions to the mKdV equations on the torus. This first enable us to construct weak solutions in $L^2$ 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-map associated with the mKdV and the KdV equation are discontinuous for the $H^s(\T)$ topology for respectively $s<0$ and $s<-1$. These last results are sharp.