Improvement of the Parabolic Regularization Method and Applications to Dispersive Models

TL;DR

This paper improves the parabolic regularization method to prove global well-posedness for the Benjamin Ono and dispersion-generalized Benjamin Ono equations in Sobolev spaces, avoiding previous gauge transformation techniques.

Contribution

It introduces a modified parabolic regularization approach that establishes global well-posedness for these dispersive models without relying on gauge transformations.

Findings

Proves global well-posedness of Benjamin Ono in H^s for s > 1/2

Extends results to dispersion-generalized Benjamin Ono equation

Avoids use of gauge transformation in the proof

Abstract

We prove that the Benjamin Ono equation is globally well-posed in $H^s(\mathbb{R})$ for $s > 1/2$. Our approach does not rely on the global gauge transformation introduced by Tao (arXiv:math/0307289). Instead, we employ a modified version of the standard parabolic regularization method. In particular, this technique also enables us to establish global well-posedness, in the same Sobolev space, for the dispersion-generalized Benjamin Ono (DGBO) equation.