Sharp well-posedness for the Chen-Lee equation

TL;DR

This paper establishes sharp well-posedness results for the Chen-Lee equation in Sobolev spaces, analyzes the flow map regularity, and investigates the asymptotic behavior of solutions as parameters tend to zero.

Contribution

It proves the sharpness of well-posedness thresholds and explores the limiting and asymptotic behaviors of solutions for the Chen-Lee equation.

Findings

Well-posedness holds for s > -1/2, but fails to be C^3 for s < -1/2.

Solutions exhibit specific limiting behaviors as dispersive and dissipative parameters approach zero.

Solutions' asymptotic behavior at infinity is characterized in weighted Sobolev spaces.

Abstract

We study the initial value problem associated to a perturbation of the Benjamin-Ono equation or Chen-Lee equation. We prove that results about local and global well-posedness for initial data in $H^s(R)$, with $s>-1/2$, are sharp in the sense that the flow-map data-solution fails to be $C^3$ in $H^s(\mathbb{R})$ when $s<-\frac{1}{2}$. Also, we determine the limiting behavior of the solutions when the dispersive and dissipative parameters goes to zero. In addition, we will discuss the asymptotic behavior (as $|x|\to \infty$) of the solutions by solving the equation in weighted Sobolev spaces.