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 $\dot{H}^s$, for $s\ge0$. 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 $\dot{H}^s$ to $\dot{H}^s$, for $s\in(-1,0]$.