| Abstract: |
| In this paper, we consider the chemotaxis-Navier-Stokes system with nonlinear diffusion $\Delta n^m$ $(m>0)$ and rotational flux given by
\begin{eqnarray*}
\left\{\begin{array}{lll}
\medskip
n_{t}+u\cdot\nabla n=\Delta n^m-\nabla\cdot(uS(x,n,c)\cdot\nabla c),&x\in\Omega,\ \ t>0,\
\medskip
c_t+u\cdot\nabla c=\Delta c+n-c,&x\in\Omega,\ \ t>0,\
\medskip
u_t+k(u\cdot\nabla)u=\Delta u+\nabla P+n\nabla\phi,&x\in\Omega,\ \ t>0\
\medskip
\nabla u_\varepsilon=0,&x\in\Omega,\ \ t>0
\end{array}\right.
\end{eqnarray*}
under homogenous Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^3$, where $k\in\mathbb{R}$, $\phi\in W^{1,\infty}(\Omega)$, the given tensor-valued function $S$: $\overline\Omega\times[0,\infty)^2\rightarrow\mathbb{R}^{3\times 3}$ satisfying
$$|S(x,n,c)|\leq S_0(n+1)^{-\alpha}\ \ {\rm for\ all}\ x\in\mathbb{R}^3,\ n\geq0,\ c\geq0.$$
Resorting to some yet weaker solution concepts and imposing no restrictions on the size of the initial data, we establish the global existence of a weak solution while assuming $m+\alpha>\frac{4}{3}$ and $m>\frac{1}{3}$, which includes both non-degenerate $(m>1)$ and the degenerate $(m<1)$ cases. |
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