We consider the Cauchy problem associated to a class of dispersive perturbations of Burgers' equations, which contains the low dispersion Benjamin-Ono equation, (also known as low dispersion fractional KdV equation), $$ \partial_tu-D_x^{ \alpha}\partial_xu=\partial_x(u^2) \, ,$$ and prove that it is locally well-posed in $H^s(\mathbb K)$, $\mathbb K=\mathbb R$ or $\mathbb T$, for $s>s_{ \alpha}$, where \begin{ equation*} s_\alpha=\begin{ cases} 1-\frac{ 3\alpha}4 & \text{ for} \quad \frac23 \le \alpha \le 1; \frac 32(1-\alpha) & \text{ for} \quad \frac13 \le \alpha \le \frac23; \frac 32-\frac{ \alpha}{ 1-\alpha} & \text{ for} \quad 0 < \alpha \le \frac13 . \end{ cases} \end{ equation*} The uniqueness is unconditional in $H^s(\mathbb K)$ for $s>\max\{ \frac12,s_{ \alpha}\}$. Moreover, we obtain \emph{ a priori} estimates for the solutions at the lower regularity threshold $s>\widetilde{ s}_\alpha$ where \begin{ equation*} \widetilde{ s}_\alpha=\begin{ cases} \frac 12-\frac \alpha 4 & \text{ for} \quad \frac23 \le \alpha \le 1; 1-\alpha & \text{ for} \quad \frac12 \le \alpha \le \frac23; \frac 32-\frac{ \alpha}{ 1-\alpha} & \text{ for} \quad 0 < \alpha \le \frac12 . \end{ cases} \end{ equation*} As a consequence of these results and of the Hamiltonian structure of the equation, we deduce global well-posedness in $H^s(\mathbb K)$ for $s>s_{ \alpha}$ when $\alpha>\frac23$, and in the energy space $H^{ \frac{ \alpha}2}(\mathbb K)$ when $\alpha>\frac45$.
Categories: math.AP