[Author: Wang, Yulan] AND [Keyword: Eventual Smoothness] : Search

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[Author: Fu, Heliang] AND [Keyword: Catalytic Activity] (1)	13 Mar 2025	Run
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articleNo Access
Global solvability and eventual smoothness in a chemotaxis-fluid system with weak logistic-type degradation
- Yulan Wang
Mathematical Models and Methods in Applied Sciences10 Jun 2020
Preview Abstract
We consider the coupled chemotaxis–Navier–Stokes system with logistic source term
$\{\begin{matrix} n_{t} + u \cdot \nabla n = Δ n - \nabla \cdot (n \nabla c) + r n - μ n^{α}, \\ c_{t} + u \cdot \nabla c = Δ c - n c, \\ u_{t} + (u \cdot \nabla) u = Δ u + \nabla P + n \nabla Φ, \\ \nabla \cdot u = 0 \end{matrix}$
in a bounded, smooth domain $Ω \subset ℝ^{3}$ , where $Φ \in W^{2, \infty} (Ω)$ and where $r \geq 0$ , $μ > 0$ and $1 < α < 2$ are given parameters. Although the degradation here is weaker than the usual quadratic case, it is proved that for any sufficiently regular initial data, the initial-value problem for this system under no-flux boundary conditions for $n$ and $c$ and homogeneous Dirichlet boundary condition for $u$ possesses at least one globally defined weak solution. And this weak solution becomes smooth after some waiting time provided $\frac{6}{5} \leq α < 2$ .

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