Eventual smoothness and stabilization in a three-dimensional Keller–Segel–Navier–Stokes system with rotational flux

Abstract

We consider the spatially 3-D version of the following Keller–Segel–Navier–Stokes system with rotational flux

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

(*)

under no-flux boundary conditions in a bounded domain $\Omega \subseteq \mathbb {R}^{3}$ with smooth boundary, where $\phi \in W^{2,\infty } (\Omega )$ and $\kappa \in \mathbb {R}$ represent the prescribed gravitational potential and the strength of nonlinear fluid convection, respectively. Here the matrix-valued function $S(x,n,c)\in C^2(\bar{\Omega }\times [0,\infty )^2 ;\mathbb {R}^{3\times 3})$ denotes the rotational effect which satisfies $|S(x,n,c)|\le C_S(1 + n)^{-\alpha }$ with some $C_S > 0$ and $\alpha \ge 0$. Compared with the signal consumption case as in chemotaxis-(Navier-)Stokes system, the quantity c of system $(*)$ is no longer a priori bounded by its initial norm in $L^\infty$, which means that we have less regularity information on c. Moreover, the tensor-valued sensitivity functions result in new mathematical difficulties, mainly linked to the fact that a chemotaxis system with such rotational fluxes thereby loses an energy-like structure. In this paper, under an explicit condition on the size of $C_S$ relative to $C_N$, we can prove that the weak solution (n, c, u) becomes smooth ultimately, and that it approaches the unique spatially homogeneous steady state $(\bar{n}_0,\bar{n}_0,0)$, where $\bar{n}_0=\frac{1}{|\Omega |}\int _{\Omega }n_0$ and $C_N$ is the best Poincaré constant. To the best of our knowledge, there are the first results on asymptotic behavior of the system.