Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity

Abstract

The Cauchy problem for the nonperiodic KdV equation is shown by the iteration method to be locally well-posed in $H^{-3/4}(\mathbb R )$. In particular, solutions are unique in the whole Banach space for the iteration. This extends the previous well-posedness result in $H^s$, $s>-3/4$ obtained by Kenig, Ponce and Vega (1996) to the limiting case, and improves the existence result in $H^{-3/4}$ given by Christ, Colliander and Tao (2003). Our result immediately yields global well-posedness for the KdV equation in $H^{-3/4}(\mathbb R )$ and for the modified KdV equation in $H^{1/4}(\mathbb R )$, combined with the argument of Colliander, Keel, Staffilani, Takaoka and Tao (2003).