Local well-posedness for a class of singular Vlasov equations

In this article we study a singular Vlasov system on the torus where the force field has the smoothness of a (fractional) derivative $D^{\alpha}$ of the density, where $\alpha>0$. We prove local well-posedness in Sobolev spaces without restriction on the data. This is in sharp contrast with the case $\alpha=0$ which is ill-posed in Sobolev spaces for general data.