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

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 $H_x^{0+}$ topology, where the Lipschitz constant only depends on the rough norm $\Vert u_0{\Vert }_{H^{-1/2+}}$ of the initial data. A similar episode has been observed in a recent paper on 1D quadratic Schrödinger equation in low-regularity setting.

arXiv commentMinor revisions in the abstract and the introduction. Added one reference item

References in corpus (1)

• Local well-posedness for quadratic Schrödinger equations in ${\mathbf{R}}^{1+1}$: a normal form approach