On the periodic Korteweg-de Vries equation: a normal form approach

TL;DR

This paper employs a normal form transformation to improve the understanding of smoothing effects for low-regularity solutions of the periodic KdV equation, establishing Lipschitz continuity of the solution map in a rough topology.

Contribution

It introduces a novel normal form approach to demonstrate Lipschitz continuity of the solution map for low-regularity periodic KdV solutions, enhancing previous smoothing results.

Findings

Solution map is Lipschitz in $H^{0+}_x$ topology

Lipschitz constant depends only on initial data norm

Improves understanding of low-regularity solution behavior

Abstract

This paper discusses an improved smoothing phenomena for low-regularity solutions of the Korteweg-de Vries (KdV) equation in the periodic settings by means of normal form transformation. As a result, the solution map from a ball on $H^{-1/2+}$ to $C_0^t ([0,T], H^{-1/2+})$ can be shown to be Lipschitz in a $H^{0+}_x$ topology, where the Lipschitz constant only depends on the rough norm $\|u_0\|_{H^{-1/2+}}$ of the initial data. A similar episode has been observed in a recent paper on 1D quadratic Schr\"odinger equation in low-regularity setting.