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

2022/01/31 by Young-Pil Choi, In-Jee Jeong, Choi, Young-Pil +1 · 1 citation
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.MP

paper · pdf · doi:10.48550/arxiv.2201.12988

23 pages, 1 figure

arxiv created 2022/01/31 · arxiv updated 2022/02/01

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 Φ= (-Δ)^-1ρ, Manev (-Δ)^-1 + (-Δ)^-\frac12, and pure Manev (-Δ)^-\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≥4. Our local well-posedness and singularity formation results extend to cases when linear diffusion and damping in velocity are present.

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• Global dynamics of damped Euler systems with exterior potentials

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