On the Well-posedness for the Chen-Lee equation in periodic Sobolev spaces

TL;DR

This paper establishes local and global well-posedness of a perturbed Chen-Lee equation in periodic Sobolev spaces for regularity levels above -1/2, and discusses ill-posedness for lower regularities.

Contribution

It proves well-posedness results for the Chen-Lee equation in Sobolev spaces with regularity above -1/2, extending understanding of the equation's mathematical properties.

Findings

Well-posedness in H^s for s > -1/2

Ill-posedness for s < -1

Global solutions exist for certain initial data

Abstract

We prove that the initial value problem associated to a perturbation of the Benjamin-Ono equation or Chen-Lee equation $u_t+uu_x+\beta \mathcal{H}u_{xx}+\eta (\mathcal{H}u_x - u_{xx})=0$, where $x\in \mathbb{T}$, $t> 0$, $\eta >0$ and $\mathcal{H}$ denotes the usual Hilbert transform, is locally and globally well-posed in the Sobolev spaces $H^s(\mathbb{T})$ for any $s>-\frac{1}{2}$. We also prove some ill-posedness issues when $s<-1$.