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

Abstract

We prove that the initial value problem associated to a perturbation of the Benjamin-Ono equation or Chen-Lee equation $u_t+u{u}_x+\beta \mathcal{H}{u}_{xx}+\eta (\mathcal{H}{u}_x-u_{xx})=0$, 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 $s<-1$.

arXiv comment15 pages