We show that the Cauchy problem associated to a class of dispersive perturbations of Burgers' equations containing the low dispersion Benjamin-Ono equation, also known as low dispersion fractional KdV equation,

$$ \partial_tu-D_x^{\alpha}\partial_xu=\partial_x^2(u) \, ,$$

is locally well-posed in $H^s(\mathbb K)$, $\mathbb K=\mathbb R$ or $\mathbb T$, for $s>s_{\alpha}=1-\frac{3\alpha}4$ when $\frac23 \le \alpha \le 1$. The result also extend to other values of $s_{\alpha}$ when $0<\alpha <\frac23$. As a consequence of this result and of the Hamiltonian structure of the equation, we obtain global well-posedness in the energy space $H^{\frac{\alpha}2}$ as soon as $\frac\alpha 2> s_\alpha$, i.e. $\alpha>\frac45$.

In the first part of the talk, I will introduce the equations, explain their connection to fluid mechanics, and review several mathematical results and open problems.

In the second part, I will give an overview of the proof which combines an energy method for strongly nonresonant dispersive equations introduced by Molinet and Vento with refined Strichartz estimates and modified energies. Moreover, we use a full symmetrisation of the modified energy both for the a priori estimate and for the estimate of the difference of two solutions. This symmetrisation allows some cancellations of the resulting symbol which are crucial to close the estimates.

This talk is based on a joint work (still in progress) with Luc Molinet (Université de Tours) and Stéphane Vento (Université Paris 13).