Global boundedness of solutions for a quasilinear chemotaxis-Stokes system with nonlinear production

In this paper we study the global boundedness of solutions to the quasilinear chemotaxis-Stokes system with nonlinear production

subject to no-flux/no-flux/Dirichlet boundary conditions in a smoothly bounded domain $$\Omega \subset \mathbb {R}^N$$ with $$N\in \lbrace 2,3\rbrace$$, where $$\beta \ge 0$$, $$\phi \in W^{2,\infty }(\Omega)$$, and $$D,S\in C^2([0,\infty))$$ generalize the prototypes $$D(s)=k_D(s+1)^{-m}$$ and $$S(s)=K_S(s+1)^{-\alpha }$$ for all $$s\ge 0$$ with $$k_D,K_S>0$$ and $$m,\alpha \in \mathbb {R}$$. It is shown that if $$\alpha >m+\beta -\frac{2}{N}$$ for $$N=2$$, or $$\alpha >m+(\beta -\frac{2}{N})_{+}$$ for $$N=3$$, then for any reasonably regular initial datum, the corresponding initial-boundary value problem of ($$\star$$) possesses a unique globally bounded classical solution. Compared to the global boundedness condition $$\alpha >m+\beta -\frac{2}{N}$$ for $$N\in \lbrace 2,3\rbrace$$ in the relevant fluid-free system [J. Differ. Equ. 268 6729–6777 (2020)], the present result indicates that the slow fluid motion described by the Stokes equation might possibly be adverse to the global solvability in the case that $$\beta <\frac{2}{N}$$ with $$N=3$$. What is more, this paper provides for the quasilinear problems with general signal production a universal approach capable of deriving the boundedness of solutions in the optimal range of parameters.