Czechoslovak Mathematical Journal, Vol. 73, No. 1, pp. 49-70, 2023


Asymptotic behavior of small-data solutions to a Keller-Segel-Navier-Stokes system with indirect signal production

Lu Yang, Xi Liu, Zhibo Hou

Received October 24, 2021.   Published online October 12, 2022.

Abstract:  We consider the Keller-Segel-Navier-Stokes system $$\begin{cases} n_t+ u\cdot\nabla n =\Delta n - \nabla\cdot(n\nabla v ),& x\in\Omega, t>0,\\ v_t + u\cdot\nabla v=\Delta v -v+w, &x\in\Omega, t>0, \\ w_t+ u\cdot\nabla w=\Delta w -w+n, &x\in\Omega, t>0,\\ {\bf{u}}_t + ({\bf{u}}\cdot\nabla){\bf{u}} = \Delta{\bf{u}} + \nabla P + n\nabla\phi, \nabla\cdot u=0, &x\in\Omega, t>0, \end{cases}$$ which is considered in bounded domain $\Omega\subset\mathbb{R}^N$ $(N \in\{2,3\})$ with smooth boundary, where $\phi\in C^{1+\delta}(\overline\Omega)$ with $\delta\in(0,1)$. We show that if the initial data $\|n_0\|_{L^{N/2}(\Omega)}$, $\|\nabla v_0\|_{L^N(\Omega)}$, $\|\nabla w_0\|_{L^N(\Omega)}$ and $\| u_0\|_{L^N(\Omega)}$ is small enough, an associated initial-boundary value problem possesses a global classical solution which decays to the constant state $({\bar n}_0,{\bar n}_0,{\bar n}_0,0)$ exponentially with ${\bar n}_0:=(1/|\Omega|)\int_{\Omega}n_0(x){\rm d}x$.
Keywords:  Keller-Segel-Navier-Stokes; global solution; decay estimate; indirect process
Classification MSC:  35K55, 35B40, 35Q35, 92C17, 35B35


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Affiliations:   Lu Yang, School of Mathematical Sciences, University of Electronic Science and Technology of China, No.2006, Xiyuan Ave. West Hi-Tech. Zone, 611731 Chengdu, P. R. China, e-mail: math_lu96@163.com; Xi Liu, Zhibo Hou (corresponding author), School of Science, Xihua University, 610039 Chengdu, P. R. China, e-mail: xhliuxi@163.com, houzhibo@mail.xhu.edu.cn


 
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