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arXiv · 2405.02557

Structure of singularities for the Euler-Poisson system of ion dynamics

Junsik Bae·Yunjoo Kim·Bongsuk Kwon
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Abstract

We study the formation of singularity for the isothermal Euler-Poisson system arising from plasma physics. Contrast to the previous studies yielding only limited information on the blow-up solutions, for instance, sufficient conditions for the blow-up and the temporal blow-up rate along the characteristic curve, we rather give a constructive proof of singularity formation from smooth initial data. More specifically, employing the stable blow-up profile of the Burgers equation in the self-similar variables, we establish the global stability estimate in the self-similar time, which yields the asymptotic behavior of blow-up solutions near the singularity point. Our analysis indicates that the smooth solution to the Euler-Poisson system can develop a cusp-type singularity; it exhibits $C^1$ blow-up in a finite time, while it belongs to $C^{1/3}$ at the blow-up time, provided that smooth initial data are sufficiently close to the blow-up profile in some weighted $C^4$-topology. We also present a similar result for the isentropic case, and discuss noteworthy differences in the analysis.

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Junsik Bae, Yunjoo Kim, Bongsuk Kwon. 2024-05-04. Structure of singularities for the Euler-Poisson system of ion dynamics. https://arxiv.org/abs/2405.02557

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