Global Bounded Solution in a Chemotaxis-Stokes Model with Porous Medium Diffusion and Singular Sensitivity

Abstract

This article is concerned with a chemotaxis-Stokes system with porous medium diffusion and singular sensitivity:

$$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l} n_{t}+u\cdot \nabla n=\nabla \cdot (D(n)\nabla n)-\nabla \cdot (nS(x,n,c)\cdot \nabla c),&x\in \Omega ,\ \ t>0, \\ c_{t}+u\cdot \nabla c=\Delta c-nc,&x\in \Omega ,\ \ t>0, \\ u_{t}+\nabla P=\Delta u+n\nabla \Phi ,\ \ \ \nabla \cdot u=0,&x\in \Omega ,\ \ t>0 \end{array}\displaystyle \right . \end{aligned}$$

in a bounded domain \(\Omega \subset \mathbb{R}^{N}\) with \(2\le N\le 3\), where \(D\in C^{0}([0,\infty ))\cap C^{2}((0,\infty ))\) and \(S\in C^{2}(\bar{\Omega }\times [0,\infty )^{2};\mathbb{R}^{N\times N})\). The global solvability of the system in a natural weak sense is obtained under the conditions that \(D(n)\ge k_{D}n^{m-1}\) and \(|S(x,n,c)|\le \frac{S_{0}(c)}{c^{\alpha }}\) for all \((x,n,c)\in \Omega \times (0,\infty )^{2}\) with some \(k_{D}>0\), \(m>\frac{3N-2}{2N}\), \(\alpha \in [0,1)\) and some nondecreasing \(S_{0}:(0,\infty )\rightarrow (0,\infty )\). Moreover, in the case that \(m=\frac{3N-2}{2N}\) and \(\alpha \in [0,1)\), we also get the global weak solutions under smallness assumptions on the initial data \(\|n_{0}\|_{L^{1}(\Omega )}\) and \(\|c_{0}\|_{L^{\infty }(\Omega )}\).