Shock-type singularity of the hyperbolic-parabolic chemotaxis system
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913653578530816 |
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| author | Lee, Woojae |
| author_facet | Lee, Woojae |
| contents | This paper deals with the hyperbolic-parabolic chemotaxis (HPC) model, which is a hydrodynamic model describing vascular network formation at the early stage of the vasculature. We study analytically the singularity formation associated with the shock-type structure, which was numerically observed by Filbet, Lauren{ç}ot, and Perthame \cite{filbet2005derivation} and Filbet and Shu \cite{filbet2005approximation}. We construct the blow-up profile in a 1D HPC system on $\mathbb{R}$ as follows: The blow-up profile is stable in the sense of $H^m$ topology ($m\geq 5$) prior to the occurrence of the singularity. For the first singularity, while the density and velocity $(ρ, u)$ of endothelial cells themselves remain bounded, the gradients of the density and velocity blow up. The chemoattractant concentration $ϕ$ has $C^2$ regularity. However, the density and velocity with $C^ {\frac{1}{3}}$ regularity exhibit a cusp singularity at a unique blow-up point, the location and time of which are explicitly estimated. Furthermore, the HPC system is $C^1$ differentiable except in any neighborhood of the blow-up point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_09656 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Shock-type singularity of the hyperbolic-parabolic chemotaxis system Lee, Woojae Analysis of PDEs This paper deals with the hyperbolic-parabolic chemotaxis (HPC) model, which is a hydrodynamic model describing vascular network formation at the early stage of the vasculature. We study analytically the singularity formation associated with the shock-type structure, which was numerically observed by Filbet, Lauren{ç}ot, and Perthame \cite{filbet2005derivation} and Filbet and Shu \cite{filbet2005approximation}. We construct the blow-up profile in a 1D HPC system on $\mathbb{R}$ as follows: The blow-up profile is stable in the sense of $H^m$ topology ($m\geq 5$) prior to the occurrence of the singularity. For the first singularity, while the density and velocity $(ρ, u)$ of endothelial cells themselves remain bounded, the gradients of the density and velocity blow up. The chemoattractant concentration $ϕ$ has $C^2$ regularity. However, the density and velocity with $C^ {\frac{1}{3}}$ regularity exhibit a cusp singularity at a unique blow-up point, the location and time of which are explicitly estimated. Furthermore, the HPC system is $C^1$ differentiable except in any neighborhood of the blow-up point. |
| title | Shock-type singularity of the hyperbolic-parabolic chemotaxis system |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.09656 |