Global existence, smooth and stabilization in a three-dimensional Keller-Segel-Navier-Stokes system with rotational flux

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Abstract

We consider the spatially $3$-D version of the following Keller-Segel-Navier-Stokes system with rotational flux $$\left\{\begin{array}{l} n_t+u\cdot\nabla n=\Delta n-\nabla\cdot(nS(x,n,c)\nabla c),\quad x\in \Omega, t>0, c_t+u\cdot\nabla c=\Delta c-c+n,\quad x\in \Omega, t>0,\\ u_t+\kappa(u \cdot \nabla)u+\nabla P=\Delta u+n\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u=0,\quad x\in \Omega, t>0 \end{array}\right.\qquad(*)$$ 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 $|S(x,n,c)|\leq C_S(1 + n)^{-\alpha}$ with some $C_S > 0$ and $\alpha\geq 0$. In this paper, by seeking some new functionals and using the bootstrap arguments on system $(*)$, we establish the existence of global weak solutions to system $(*)$ for arbitrarily large initial data under the assumption $\alpha\geq1$. Moreover, under an explicit condition on the size of $C_S$ relative to $C_N$, we can secondly prove that in fact any such {\bf 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 Poincar\'{e} constant. To the best of our knowledge, there are the first results on asymptotic behavior of the system.