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

Abstract

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:=32−5α4. As a consequence, we obtain global well-posedness in the energy space Hα2(R) as soon as α2>sα, i.e. α>67.