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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-s initial data, s > -1/2, and for any s(1) = 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-2 theory of Bourgain) is a continuous function of space and time. In addition, we demonstrate smoothing for the defocusing modified KdV equation on the torus for s > 1/2.