Local well-posedness for a class of singular Vlasov equations

Abstract

<p style='text-indent:20px;'>In this article we study a singular Vlasov system on the torus where the force field has the smoothness of a (fractional) derivative <inline-formula><tex-math id="M1">\begin{document}$ D^{\alpha} $\end{document}</tex-math></inline-formula> of the density, where <inline-formula><tex-math id="M2">\begin{document}$ \alpha>0 $\end{document}</tex-math></inline-formula>. We prove local well-posedness in Sobolev spaces without restriction on the data. This is in sharp contrast with the case <inline-formula><tex-math id="M3">\begin{document}$ \alpha = 0 $\end{document}</tex-math></inline-formula> which is ill-posed in Sobolev spaces for general data.</p>