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

TL;DR

This paper investigates the regularization mechanism of the periodic KdV equation using averaging methods, establishing global well-posedness and continuous dependence of solutions in various Sobolev spaces.

Contribution

It introduces successive averaging techniques to explain regularization in the periodic KdV and proves well-posedness and stability results in Sobolev spaces.

Findings

Proves global existence and uniqueness of solutions in $ ext{H}^s$ for $s \uge 0$.

Establishes Lipschitz continuous dependence on initial data in Sobolev spaces.

Demonstrates continuous dependence from $ ext{H}^s$ to $ ext{H}^s$ for $s r(-1,0]$.

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