The IVP for a nonlocal perturbation of the Benjamin-Ono equation in classical and weighted Sobolev spaces

TL;DR

This paper establishes local and global well-posedness for a nonlocal perturbation of the Benjamin-Ono equation in Sobolev spaces, identifies the sharp regularity threshold, and explores persistence and unique continuation in weighted spaces.

Contribution

It proves well-posedness results for the perturbed Benjamin-Ono equation in Sobolev spaces and analyzes the regularity threshold and properties in weighted Sobolev spaces.

Findings

Well-posedness for s > -3/2 in Sobolev spaces.

Flow map not C^2 for s < -3/2, indicating sharpness.

Persistence and unique continuation in weighted Sobolev spaces.

Abstract

We prove that the initial value problem associated to a nonlocal perturbation of the Benjamin-Ono equation is locally and globally well-posed in Sobolev spaces $H^s(\mathbb{R})$ for any $s>-3/2$ and we establish that our result is sharp in the sense that the flow map of this equation fails to be $C^2$ in $H^s(\mathbb{R})$ for $s<-3/2$. Finally, we study persistence properties of the solution flow in the weighted Sobolev spaces $Z_{s,r}=H^s(\mathbb{R})\cap L^2(|x|^{2r}\,dx)$ for $s\geq r >0$. We also prove some unique continuation properties of the solution flow in these spaces.