Delta-shock for the pressureless Euler-Poisson system
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| Format: | Preprint |
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2024
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| author | Bae, Junsik Kim, Yunjoo Kwon, Bongsuk |
| author_facet | Bae, Junsik Kim, Yunjoo Kwon, Bongsuk |
| contents | We study singularity formation for the pressureless Euler-Poisson system of cold ion dynamics. In contrast to the Euler-Poisson system with pressure, when its smooth solutions experience $C^1$ blow-up, the $L^\infty$ norm of the density becomes unbounded, which is often referred to as a delta-shock. We provide a constructive proof of singularity formation to obtain an exact blow-up profile and the detailed asymptotic behavior of the solutions near the blow-up point in both time and space. Our result indicates that at the blow-up time $t=T_\ast$, the density function is unbounded but is locally integrable with the profile of $ρ(x,T_\ast) \sim (x-x_*)^{-2/3}$ near the blow-up point $x=x_\ast$. This profile is not yet a Dirac measure. On the other hand, the velocity function has $C^{1/3}$ regularity at the blow-up point. Loosely following our analysis, we also obtain an exact blow-up profile for the pressureless Euler equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_15669 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Delta-shock for the pressureless Euler-Poisson system Bae, Junsik Kim, Yunjoo Kwon, Bongsuk Analysis of PDEs We study singularity formation for the pressureless Euler-Poisson system of cold ion dynamics. In contrast to the Euler-Poisson system with pressure, when its smooth solutions experience $C^1$ blow-up, the $L^\infty$ norm of the density becomes unbounded, which is often referred to as a delta-shock. We provide a constructive proof of singularity formation to obtain an exact blow-up profile and the detailed asymptotic behavior of the solutions near the blow-up point in both time and space. Our result indicates that at the blow-up time $t=T_\ast$, the density function is unbounded but is locally integrable with the profile of $ρ(x,T_\ast) \sim (x-x_*)^{-2/3}$ near the blow-up point $x=x_\ast$. This profile is not yet a Dirac measure. On the other hand, the velocity function has $C^{1/3}$ regularity at the blow-up point. Loosely following our analysis, we also obtain an exact blow-up profile for the pressureless Euler equations. |
| title | Delta-shock for the pressureless Euler-Poisson system |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.15669 |