Wellposedness for the KdV hierarchy

Abstract

We prove a version of wellposedness for all equations of the KdV hierarchy in $H^{-1}$. Ingredients are 1) The Miura map which allows to define the Gardner hierarchy through the generating function of the energies so that the $N$th Gardner equation is equivalent to the $N$th KdV equation. 2) A rigorous relation between the generating functions of the energies and the KdV resp. Gardner Hamiltonians. 3) Kato smoothing estimates for weak solutions and approximate flows. Section 2 has been rewritten. Typos corrected-