WELL-POSEDNESS OF THE CAUCHY PROBLEM FOR THE KORTEWEG-DE VRIES EQUATION AT THE CRITICAL REGULARITY

The Cauchy problem for the nonperiodic KdV equation is shown by the iteration method to be locally well-posed in H(-3/4)(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)(R) and for the modified KdV equation in H(1/4)(R), combined with the argument of Colliander, Keel, Staffilani, Takaoka and Tao (2003).