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 ∂ ₜu−Dₓα∂ ₓu = ∂ ₓ(u²)\:,\:0 < α ≤ 1, is locally well-posed in Hˢ(R) when s > sα: = 3/2−5α /4 . As a consequence, we obtain global well-posedness in the energy space Hα /2(R) as soon as α /2 > sα , i.e. α > 6/7 .