Well-posedness and singularity formation for Vlasov--Riesz system

Abstract

We investigate the Cauchy problem for the Vlasov--Riesz system, which is a Vlasov equation featuring an interaction potential generalizing previously studied cases, including the Coulomb $\Phi = (-\Delta)^{-1}\rho$, Manev $(-\Delta)^{-1} + (-\Delta)^{-\frac12}$, and pure Manev $(-\Delta)^{-\frac12}$ potentials. For the first time, we extend the local theory of classical solutions to potentials more singular than that for the Manev. Then, we obtain finite-time singularity formation for solutions with various attractive interaction potentials, extending the well-known blow-up result of Horst for attractive Vlasov--Poisson for $d\ge4$. Our local well-posedness and singularity formation results extend to cases when linear diffusion and damping in velocity are present.