  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.
