Global Smoothing for the Periodic KdV Evolution

Abstract

The Korteweg–de Vries (KdV) equation with periodic boundary conditions is considered. It is shown that for Hs initial data, ⁠, and for any ⁠, the difference of the nonlinear and linear evolutions is in Hs1 for all times, with at most polynomially growing Hs1 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 L2 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 ⁠.