# On well-posedness for some dispersive perturbations of Burgers' equation

## TL;DR

This paper establishes local and global well-posedness results for dispersive perturbations of Burgers' equation, including the Benjamin-Ono equation, in specific Sobolev spaces, depending on the dispersion parameter.

## Contribution

It proves well-posedness for a class of dispersive Burgers' equations, extending known results to low dispersion regimes and identifying conditions for global solutions.

## Key findings

- Local well-posedness in H^s for s > 3/2 - 5α/4
- Global well-posedness in H^{α/2} for α > 6/7
- Includes the low dispersion Benjamin-Ono equation case

## Abstract

We show that the Cauchy problem for a class of dispersive perturbations of Burgers' equations containing the low dispersion Benjamin-Ono equation $\partial$\_t u -- D^$\alpha$\_x $\partial$\_x u = $\partial$\_x(u^2), 0 < $\alpha$ $\le$ 1, is locally well-posed in H^s (R) when s > 3 /2 -- 5$\alpha$ /4. As a consequence, we obtain global well-posedness in the energy space H^{$\alpha$/2} (R) as soon as $\alpha$ > 6/7 .