Global Smoothing for the Periodic Kdv Evolution

The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. It is shown that for H initial data, s > −1/2, and for any s1 < min(3s+1, s+ 1), the difference of the nonlinear and linear evolutions is in H1 for all times, with at most polynomially growing H1 norm. The result also extends to KdV with a smooth, mean zero, time-dependent potential in the case s ≥ 0. Our result and a theorem of Oskolkov for the Airy evolution imply that if one starts with continuous and bounded variation initial data then the solution of KdV (given by the L theory of Bourgain) is a continuous function of space and time. In addition, we demonstrate smoothing for the modified KdV equation on the torus for s > 1/2.