Global well-posedness of the Benjamin-Ono equation in H^1(R)

TL;DR

This paper proves the global well-posedness of the Benjamin-Ono equation in the Sobolev space H^1(R) by employing a novel gauge transformation to handle the derivative non-linearity.

Contribution

It introduces a global gauge transformation technique that effectively manages the derivative in the non-linearity, establishing well-posedness in H^1(R).

Findings

Global well-posedness in H^1(R) established.

The gauge transformation reduces the impact of the derivative non-linearity.

Solution map is not uniformly continuous in H^s for any s.

Abstract

We show that the Benjamin-Ono equation is globally well-posed in $H^s(\R)$ for $s \geq 1$. This is despite the presence of the derivative in the non-linearity, which causes the solution map to not be uniformly continuous in $H^s$ for any $s$. The main new ingredient is to perform a global gauge transformation which almost entirely eliminates this derivative.