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

Corrected version accepted for publication in Annales de l'IHP.We show that the Cauchy problem for a class of dispersive perturbations of Burgers' equations containing the low dispersion Benjamin-Ono equation ∂_t u − D^α_x ∂_x u = ∂_x(u^2) , 0 3 /2 − 5α /4. As a consequence, we obtain global well-posedness in the energy space H^{α/2} (R) as soon as α > 6/7