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

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 .