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Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux

  • * Corresponding author

    * Corresponding author
The authors are supported in part by NSF of China (No. 11671079, No. 11701290, No. 11601127, No. 11871147 and No. 11871148), and NSF of Jiangsu Province (No. BK20170896).
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  • In this paper, we consider the chemotaxis-Navier-Stokes system with nonlinear diffusion and rotational flux given by

    $\begin{eqnarray*} \left\{\begin{array}{lll}n_{t}+u\cdot\nabla n=\Delta n^m-\nabla\cdot(uS(x,n,c)\cdot\nabla c),&x\in\Omega,\ \ t>0,\\[1mm]c_t+u\cdot\nabla c=\Delta c-c+n,&x\in\Omega,\ \ t>0,\\[1mm]u_t+k(u\cdot\nabla)u=\Delta u+\nabla P+n\nabla\phi,&x\in\Omega,\ \ t>0\\[1mm]\nabla\cdot u=0,&x\in\Omega,\ \ t>0 \end{array}\right.\end{eqnarray*}$

    in a bounded domain $\Omega\subset\mathbb{R}^3$, where $k\in\mathbb{R}$, $\phi\in W^{2,\infty}(\Omega)$ and the given tensor-valued function $S$: $\overline\Omega\times[0,\infty)^2\rightarrow\mathbb{R}^{3\times 3}$ satisfies

    $|S(x,n,c)|\leq S_0(n+1)^{-\alpha}\ \ {\rm for\ all}\ x\in\mathbb{R}^3,\ n\geq0,\ c\geq0.$

    Imposing no restriction on the size of the initial data, we establish the global existence of a very weak solution while assuming $m+\alpha>\frac{4}{3}$ and $m>\frac{1}{3}$.

    Mathematics Subject Classification: 35K65, 35Q35, 35Q51, 92C17.


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