Well-posedness of the periodic dispersion-generalized Benjamin-Ono equation in the weakly dispersive regime

We study the dispersion-generalized Benjamin-Ono equation in the periodic setting. This equation interpolates between the Benjamin-Ono equation ($\alpha =1$) and the viscous Burgers’ equation ($\alpha =0$). We obtain local well-posedness in $H^s$ for $s>3\mathrm{/}2-\alpha$ and $\alpha \in (0,1)$ by using the short-time Fourier restriction method.