Hele-Shaw limit of chemotaxis-Navier-Stokes flows
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913892097064960 |
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| author | He, Qingyou Shou, Ling-Yun Wu, Leyun |
| author_facet | He, Qingyou Shou, Ling-Yun Wu, Leyun |
| contents | This paper investigates the connection between the chemotaxis--Navier--Stokes system with porous medium type nonlinear diffusion and the Hele--Shaw problem in $\mathbb{R}^d$ ($d\geq2$). First, we prove the global-in-time existence of weak solutions for the Cauchy problem of the chemotaxis-Navier-Stokes system with the general initial data, uniformly in the diffusion range $m\in [3,\infty)$. Then, we rigorously justify the Hele--Shaw limit for this system as $m\rightarrow\infty$, showing the convergence to a free boundary problem of Hele--Shaw type, where the bacterium (cell) diffusion is governed by the stiff pressure law. Moreover, the complementarity relation characterizing the limiting bacterium (cell) pressure via a degenerate elliptic equation is verified by a novel application of the Hele--Shaw framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_11757 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hele-Shaw limit of chemotaxis-Navier-Stokes flows He, Qingyou Shou, Ling-Yun Wu, Leyun Analysis of PDEs This paper investigates the connection between the chemotaxis--Navier--Stokes system with porous medium type nonlinear diffusion and the Hele--Shaw problem in $\mathbb{R}^d$ ($d\geq2$). First, we prove the global-in-time existence of weak solutions for the Cauchy problem of the chemotaxis-Navier-Stokes system with the general initial data, uniformly in the diffusion range $m\in [3,\infty)$. Then, we rigorously justify the Hele--Shaw limit for this system as $m\rightarrow\infty$, showing the convergence to a free boundary problem of Hele--Shaw type, where the bacterium (cell) diffusion is governed by the stiff pressure law. Moreover, the complementarity relation characterizing the limiting bacterium (cell) pressure via a degenerate elliptic equation is verified by a novel application of the Hele--Shaw framework. |
| title | Hele-Shaw limit of chemotaxis-Navier-Stokes flows |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.11757 |