This talk is based on a joint work with Luc Molinet (Universite de Tours) and Stephane Vento (Universite Paris 13)

We show that the Cauchy problem associated to the fractional KdV equation

$$ \partial_tu-D_x^{\alpha}\partial_xu+u\partial_xu=0 \, ,$$

with low dispersion $0<\alpha \le 1$, is locally well-posed in $H^s(\mathbb R)$ for $s>s_\alpha: = \frac 32-\frac {5\alpha} 4$.

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. $\alpha>\frac67$.