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

TL;DR

This paper studies the well-posedness and finite-time singularity formation in the Vlasov--Riesz system, a generalized Vlasov equation with singular interaction potentials, extending previous results to more singular cases and including effects of diffusion and damping.

Contribution

It extends local solution theory to more singular potentials and establishes finite-time blow-up results for a broader class of Vlasov systems, including those with diffusion and damping.

Findings

Established local well-posedness for more singular potentials.

Proved finite-time singularity formation for attractive interactions.

Extended results to systems with diffusion and damping effects.

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.