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

Abstract

Abstract 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 Ḣ 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 Ḣ s to Ḣ s for s ∈(−1,0]. © 2010 Wiley Periodicals, Inc.