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

is locally well-posed in $H^s(\mathbb{R})$ when $s>s_{\alpha }:=\frac{3}{2}-\frac{5\alpha }{4}$. As a consequence, we obtain global well-posedness in the energy space $H^{\frac{\alpha }{2}}(\mathbb{R})$ as soon as $\frac{\alpha }{2}>s_{\alpha }$, i.e. $\alpha >\frac{6}{7}$.