Boundedness of weak solutions to a 3D chemotaxis-Stokes system with slow p−Laplacian diffusion and rotation

Abstract

In this paper, we consider the following chemotaxis-Stokes system with general sensitivity and rotation $\left(\begin{array}{cc}{n}_t+u\cdot \nabla n=\nabla \cdot ({|\nabla n|}^{p-2}\nabla n)-\nabla \cdot (nS(x,n,c)\nabla c),& x\in \Omega ,t>0,\\ c_t+u\cdot \nabla c=\Delta c-nc,& x\in \Omega ,t>0,\\ u_t+\nabla P=\Delta u+n\nabla \varphi ,& x\in \Omega ,t>0,\\ \nabla \cdot u=0,& x\in \Omega ,t>0\end{array}\right)$in a smooth bounded domain $\Omega \in R^3$ with zero-flux boundary and no-slip boundary condition. We prove the boundedness of the weak solutions to the initial–boundary value problem of the 3D chemotaxis-Stokes system with $p-$Laplacian diffusion if $\frac{4}{3}p+\alpha >\frac{7}{2}-\frac{\sqrt{6}}{3}$ and $p>2$, which improves the results of papers [Chen et al., Nonlinear Anal. Real World Appl., 76 (2024) 103996; Zhuang et al., Nonlinear Anal. Real World Appl., 56 (2020) 103163 and Tao et al., J. Differ. Equ., 268(11) (2020) 6879–6919] and extends the result of the paper [Jin, J. Differ. Equ. 287 (2021) 148–184] to the chemotaxis system with general sensitivity and rotation.