Global existence and boundedness in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity

In this paper, we consider the following chemotaxis-Stokes system with nonlinear diffusion and rotational flux CNFnt+u·∇n=Δnm-∇·(nS(x,n,c)·∇c),x∈Ω,t>0,ct+u·∇c=Δc-nc,x∈Ω,t>0,ut+∇P=Δu+n∇ϕ,x∈Ω,t>0,∇·u=0,x∈Ω,t>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{l} n_t+u\cdot \nabla n=\Delta n^m-\nabla \cdot (nS(x,n,c)\cdot \nabla c),\quad x\in \Omega , t>0,\\ c_t+u\cdot \nabla c=\Delta c-nc,\quad x\in \Omega , t>0,\\ u_t+\nabla P=\Delta u+n\nabla \phi ,\quad x\in \Omega , t>0,\\ \nabla \cdot u=0,\quad x\in \Omega , t>0\\ \end{array}\right. \end{aligned}$$\end{document}in a bounded domain Ω⊆R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subseteq \mathbb {R}^3$$\end{document} with smooth boundary, which describes the motion of oxygen-driven swimming bacteria in an incompressible fluid. Here the matrix-valued function S∈C2(Ω¯×[0,∞)2;R3×3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S\in C^2(\bar{\Omega }\times [0,\infty )^2;\mathbb {R}^{3\times 3})$$\end{document} fulfills |S(x,n,c)|≤S0(c)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|S(x,n,c)| \le S_0(c)$$\end{document} for all (x,n,c)∈Ω¯×[0,∞)×[0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x,n,c)\in \bar{\Omega } \times [0, \infty )\times [0, \infty )$$\end{document} with S0(c)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_0(c)$$\end{document} nondecreasing on [0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,\infty )$$\end{document}. With developing some new methods (see Sect. 4 and Sect. 5), it is proved that under the condition m>109\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m>\frac{10}{9}$$\end{document} and proper regularity hypotheses on the initial data, the corresponding initial-boundary problem possesses at least one global weak solution, which is uniformly bounded. In view of S is a tensor-valued chemotactic sensitivity, it is easy to see that the restriction on m here is optimal (see Remark 3.1) and thus answer the open problem left in Bellomo–Belloquid–Tao–Winkler (Bellomo, N., Belloquid, A., Tao, Y., Winkler, M.: toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues. Math. Models Methods Appl. Sci. 25, 1663–1763 (2015)) and Tao–Winkler (Tao, Y., Winkler, M.: locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion. Ann. Inst. H. Poincaré Anal. Non Linéaire 30, 157–178 (2013)). This result significantly improves or extends previous results of several authors (see Remark 1.1).