Global well-posedness for Vlasov equations with power-law interactions in $d\ge 4$

We study the Vlasov equation with general power-law (Riesz-type) potentials with exponent $\alpha$. We prove global well-posedness in every dimension $d\ge 4$, for both attractive and repulsive interactions, throughout the range $0<\alpha <3$, for arbitrary nonnegative compactly supported bounded initial data. No smallness condition is imposed on the initial data. For the attractive problem, the virial identity gives finite-time breakdown for negative-energy solutions when $\alpha \ge 3$. The proof of global well-posedness combines the analysis of characteristics, estimates in Lagrangian coordinates, and a crucial application of a \emph{compensated integrability} inequality due to Denis Serre.