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arXiv · 2201.12988

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

Young-Pil Choi·In-Jee Jeong
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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.

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Young-Pil Choi, In-Jee Jeong. 2022-01-31. Well-posedness and singularity formation for Vlasov--Riesz system. https://arxiv.org/abs/2201.12988

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