Global Existence, Regularity and Boundedness in a Higher-dimensional Chemotaxis-Navier-Stokes System with Nonlinear Diffusion and General Sensitivity

Abstract

We consider an incompressible chemotaxis-Navier-Stokes system with nonlinear diffusion and rotational flux

$$\begin{aligned} \left\{ \begin{array}{l} n_t+u\cdot \nabla n=\Delta n^m-\nabla \cdot (nS(x,n,c)\cdot \nabla c),\quad x\in \Omega , t>0,\\ c_t+u\cdot \nabla c=\Delta c-nc,\quad x\in \Omega , t>0,\\ u_t+\kappa (u \cdot \nabla )u+\nabla P=\Delta u+n\nabla \phi ,\quad x\in \Omega , t>0,\\ \nabla \cdot u=0,\quad x\in \Omega , t>0\\ \end{array}\right. \end{aligned}$$

in a bounded domain $\Omega \subset {\mathbb {R}}^N(N=2,3)$ with smooth boundary $\partial \Omega$, where $\kappa \in {\mathbb {R}}$. The chemotaxtic sensitivity S is a given tensor-valued function fulfilling $|S(x,n,c)| \le S_0(c)$ for all $(x,n,c)\in {\bar{\Omega }} \times [0, \infty )\times [0, \infty )$ with $S_0(c)$ nondecreasing on $[0,\infty )$. By introducing some new methods (see Sect. 4 and Sect. 5), we prove that under the condition $m >1$ and some other proper regularity hypotheses on initial data, the corresponding initial-boundary problem possesses at least one global weak solution. The present work also shows that the weak solution could be bounded provided that $N= 2$. Since S is tensor-valued, it is easy to see that the restriction on m here is optimal, which answers the left question in Bellomo-Belloquid-Tao-Winkler (Math Models Methods Appl Sci 25:1663–1763, 2015) and Tao-Winkler (Ann Inst H Poincaré Anal Non Linéaire 30:157–178, 2013). And obviously, this work improves previous results of several other authors (see Remark 1.1).