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 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.

Locations

• Kinetic and Related Models - View - PDF
• arXiv (Cornell University) - View - PDF