On the regularization mechanism for the periodic Korteweg-de Vries equation

In this paper we develop and use successive averaging methods for explaining the regularization mechanism in the the periodic Korteweg–de Vries (KdV) equation in the homogeneous Sobolev spaces Ḣs for s ≥ 0. Specifically, we prove the global existence, uniqueness, and Lipschitz‐continuous dependence on the initial data of the solutions of the periodic KdV. For the case where the initial data is in L2 we also show the Lipschitz‐continuous dependence of these solutions with respect to the initial data as maps from Ḣs to Ḣs for s ∈(−1,0]. © 2010 Wiley Periodicals, Inc.