# Communications in Mathematical Sciences

## Volume 21 (2023)

### Boundedness in a three-dimensional chemotaxis-Stokes system involving a subcritical sensitivity and indirect signal production

Pages: 1589 – 1607

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n6.a7

#### Authors

Guoqiang Ren (School of Mathematics & Statistics, Huazhong University of Science & Technology, Wuhan, Hubei, China; and Hubei Key Laboratory of Engineering Modeling & Scientific Computing, Huazhong University of Science & Technology, Wuhan, Hubei, China)

Bin Liu (School of Mathematics & Statistics, Huazhong University of Science & Technology, Wuhan, Hubei, China; and Hubei Key Laboratory of Engineering Modeling & Scientific Computing, Huazhong University of Science & Technology, Wuhan, Hubei, China)

#### Abstract

This paper is concerned with the Keller–Segel–Stokes system$$\left\{ \begin{array}{lll}n_t+\textbf{u}\cdot\nabla n=\nabla\cdot(D(n)n)-\nabla\cdot(S(n)\nabla v), \\[0.2cm]v_t+\textbf{u}\cdot\nabla v=\Delta v-v+w, \\[0.2cm]w_t+\textbf{u}\cdot\nabla w=\Delta w-w+n, \\[0.2cm]\textbf{u}_t =\Delta \textbf{u}+\nabla P+n\nabla\phi,\ \ \ \nabla\cdot u=0,\end{array}\right.(\ast)$$under no-flux/no-flux/no-flux/Dirichlet boundary conditions in smoothly bounded domains $\Omega \subset \mathbb{R}^3$, with given suitably regular functions $D$, $S$ and $\varphi$. Under the assumption that there exist $m^0 \in \mathbb{R}$, $m \geq m_0$, $k_D \gt 0$ and $K_D \gt 0$ such that$$k_Ds^{m_0-1}\leq D(s)\leq K_Ds^{m-1}\ \ \textrm{for all} \ s>1,$$and that $S(0)=0$ as well as$$\frac{|S(s)|}{D(s)}\leq K_0s^\alpha\ \ \textrm{for all} \ s>1$$with $K_0 \gt 0$, it is shown that for all suitably regular initial data an associated initial-boundary value problem $(\ast)$ possesses a globally defined bounded classical solution provided $\alpha \lt \frac{8}{9}$. We underline that the same results were established for the corresponding system with direct signal production in a well-known result for $\alpha \lt \frac{2}{3}$ in $[\href{https://doi.org/10.1007/s00033-020-1285-x}{2}]$ and $[\href{https://doi.org/10.1016/j.aml.2020.106785}{51}]$. Our result rigorously confirms that the indirect signal production mechanism genuinely contributes to the global solvability of the three-dimensional Keller–Segel–Stokes system.

#### Keywords

Keller–Segel–Stokes, blow-up prevention, indirect signal production

#### 2010 Mathematics Subject Classification

35K65, 35Q92, 92C17

Guoqiang Ren was supported by NSFC (No. 12001214). Bin Liu was partially supported by NSFC (No. 12231008).