Wellposedness for the KdV hierarchy

TL;DR

This paper establishes wellposedness results for all equations in the KdV hierarchy within the Sobolev space $H^{-1}$, utilizing the Miura map, energy generating functions, and Kato smoothing estimates.

Contribution

It introduces a comprehensive framework connecting the Gardner and KdV hierarchies, extending wellposedness to negative Sobolev spaces.

Findings

Proves wellposedness for all KdV hierarchy equations in $H^{-1}$.

Establishes a rigorous relation between energy generating functions and Hamiltonians.

Utilizes Kato smoothing estimates for weak solutions and approximate flows.

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-