Global boundedness in a two-dimensional chemotaxis-Navier–Stokes system with double chemical signals and nonlinear diffusion

Abstract

This paper investigates the following chemotaxis-Navier–Stokes system with double chemical signals and nonlinear diffusion $\left(\begin{array}{ccc}{n}_t+u\cdot \nabla n=\nabla \cdot (D\left(n\right)\nabla n)-\chi \nabla \cdot (n\nabla c)+\xi \nabla \cdot (n\nabla v),& & x\in \Omega ,t>0,\\ c_t+u\cdot \nabla c=\Delta c-nc,& & x\in \Omega ,t>0,\\ v_t+u\cdot \nabla v=\Delta v-v+n,& & x\in \Omega ,t>0,\\ u_t+\kappa (u\cdot \nabla )u=\Delta u+\nabla P+n\nabla \Phi ,& & x\in \Omega ,t>0,\\ \nabla \cdot u=0,& & x\in \Omega ,t>0\end{array}\right)$in a smooth bounded domain $\Omega \subset R^2$ with no-flux/no-flux/no-flux/no-slip boundary conditions, where $\chi >0,\xi <0$ and $\kappa \in R$ are given constants. $D$ is a given function satisfying $D\left(s\right)\ge {(s+1)}^{m-1}\text{for all}s\ge 0.$We obtain the boundedness of the classical solution to the initial–boundary value problem of the 2D chemotaxis-Navier–Stokes system if $m>1$, which implies that saturation effect can prevent blow-up arising from the chemotaxis-Navier–Stokes system without any smallness condition on the initial mass.