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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.05054 |
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Table of Contents:
- This paper is concerned with the singular chemotaxis-fluid system with indirect nutrient consumption: $ n_{t}+u\cdot\nabla n=Δn-\nabla\cdot(n S(x,n,v)\cdot \nabla v);\ v_{t}+u\cdot\nabla v=Δv-vw;\ w_{t}+u\cdot\nabla w=Δw-w+n;\ u_t+(u\cdot\nabla) u=Δu-\nabla P+n\nablaΦ;\ \nabla\cdot u=0\ $ in a smooth bounded domain $Ω\subset\mathbb{R}^2$ under no-flux/Neumann/Neumann/Dirichlet boundary conditions, where $Φ\in W^{2,\infty}(Ω)$, and $S: \overlineΩ\times [0,\infty) \times (0,\infty)\rightarrow\mathbb{R}^{2\times 2}$ is a suitably smooth function that satisfies $|S(x,n,v)|\leq S_0(v) /v $ for all $(x,n,v) \in Ω\times (0,\infty)^2$ with some nondecreasing $S_0: (0,\infty)\rightarrow(0,\infty)$. For all reasonably regular initial data with a smallness assumption merely involving the quantity $\int_Ωn_0$, it is shown that the problem possesses a globally bounded classical solution, which, inter alia, exponentially stabilizes toward the spatially homogeneous state $( \frac{1}{|Ω|}\int_Ωn_0,0,\frac{1}{|Ω|}\int_Ωn_0,0)$ with respect to the norm in $L^\infty(Ω)$. This rigorously confirms that, at least in the two-dimensional setting, in comparison to the direct mechanism of nutrient consumption, an indirect mechanism can induce much more regularity of solutions to the chemotaxis--fluid system even with a singular tensor-valued sensitivity.