Local well-posedness for a class of singular Vlasov equations

TL;DR

This paper establishes local well-posedness for a class of singular Vlasov equations with fractional derivative force fields, extending the understanding of well-posedness beyond the classical case.

Contribution

It proves local well-posedness in Sobolev spaces for a singular Vlasov system with fractional force derivatives, unlike the ill-posed classical case.

Findings

Proves local well-posedness for fractional derivative Vlasov equations.

Shows the contrast with the classical case where the system is ill-posed.

Extends the mathematical understanding of singular kinetic equations.

Abstract

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.