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 (H) over dot(s) 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 L(2) we also show the Lipschitz-continuous dependence of these solutions with respect to the initial data as maps from (H) over dot(s) to (H) over dot(s) for s is an element of (-1; 0]. (C) 2010 Wiley Periodicals, Inc.