# Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation

## TL;DR

This paper investigates the long-term behavior of small localized solutions to the Benjamin-Ono equation, demonstrating nearly global dispersive dynamics and providing a simplified approach to its $L^2$ theory.

## Contribution

It establishes nearly global dispersive decay for small data solutions and offers a new, simplified proof of the $L^2$ well-posedness for the Benjamin-Ono equation.

## Key findings

- Solutions exhibit dispersive decay almost globally in time.
- Provides a simplified, self-contained proof of $L^2$ well-posedness.
- Enhances understanding of long-time dynamics for the Benjamin-Ono equation.

## Abstract

This article represents a first step toward understanding the long time dynamics of solutions for the Benjamin-Ono equation. While this problem is known to be both completely integrable and globally well-posed in $L^2$, much less seems to be known concerning its long time dynamics. Here, we prove that for small localized data the solutions have (nearly) dispersive dynamics almost globally in time. An additional objective is to revisit the $L^2$ theory for the Benjamin-Ono equation and provide a simpler, self-contained approach.