Global dynamics for the generalized chemotaxis-Navier-Stokes system in $\mathbb{R}^3$
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arXiv
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| Format: | Preprint |
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2024
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| author | He, Qingyou Shou, Ling-Yun Wu, Leyun |
| author_facet | He, Qingyou Shou, Ling-Yun Wu, Leyun |
| contents | We consider the chemotaxis-Navier-Stokes system with generalized fluid dissipation in $\mathbb{R}^3$:
\begin{eqnarray*}
\begin{cases} \partial_t n+u\cdot \nabla n=Δn- \nabla \cdot (χ(c)n \nabla c),\\ \partial_t c+u \cdot \nabla c=Δc-nf(c),\\ \partial_t u +u \cdot \nabla u+\nabla P=-(-Δ)^αu-n\nabla ϕ,\\ \nabla \cdot u=0,
\end{cases} \end{eqnarray*} which describes the motion of swimming bacteria or bacillus subtilis suspended to water flows. First, we prove some blow-up criteria of strong solutions to the Cauchy problem, including the Prodi-Serrin type criterion ($α>\frac{3}{4}$) and the Beir${\rm\tilde{a}}$o da Veiga type criterion $(α>\frac{1}{2})$. Then, we verify the global existence and uniqueness of strong solutions for arbitrarily large initial fluid velocity and bacteria density for $α\geq \frac{5}{4}$. Furthermore, in the scenario of $\frac{3}{4}<α<\frac{5}{4}$, we establish uniform regularity estimates and optimal time-decay rates of global solutions if the $L^2$-norm of initial data is small. To our knowledge, this is the first result concerning the global existence and large-time behavior of strong solutions for the chemotaxis-Navier-Stokes equations with possibly large oscillations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04498 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global dynamics for the generalized chemotaxis-Navier-Stokes system in $\mathbb{R}^3$ He, Qingyou Shou, Ling-Yun Wu, Leyun Analysis of PDEs We consider the chemotaxis-Navier-Stokes system with generalized fluid dissipation in $\mathbb{R}^3$: \begin{eqnarray*} \begin{cases} \partial_t n+u\cdot \nabla n=Δn- \nabla \cdot (χ(c)n \nabla c),\\ \partial_t c+u \cdot \nabla c=Δc-nf(c),\\ \partial_t u +u \cdot \nabla u+\nabla P=-(-Δ)^αu-n\nabla ϕ,\\ \nabla \cdot u=0, \end{cases} \end{eqnarray*} which describes the motion of swimming bacteria or bacillus subtilis suspended to water flows. First, we prove some blow-up criteria of strong solutions to the Cauchy problem, including the Prodi-Serrin type criterion ($α>\frac{3}{4}$) and the Beir${\rm\tilde{a}}$o da Veiga type criterion $(α>\frac{1}{2})$. Then, we verify the global existence and uniqueness of strong solutions for arbitrarily large initial fluid velocity and bacteria density for $α\geq \frac{5}{4}$. Furthermore, in the scenario of $\frac{3}{4}<α<\frac{5}{4}$, we establish uniform regularity estimates and optimal time-decay rates of global solutions if the $L^2$-norm of initial data is small. To our knowledge, this is the first result concerning the global existence and large-time behavior of strong solutions for the chemotaxis-Navier-Stokes equations with possibly large oscillations. |
| title | Global dynamics for the generalized chemotaxis-Navier-Stokes system in $\mathbb{R}^3$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.04498 |