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An optimal result for global existence and boundedness in a three-dimensional Keller-Segel-Stokes system with nonlinear diffusion. (arXiv:1807.01156v1 [math.AP])

来源于:arXiv
This paper investigates the following quasilinear Keller-Segel-Navier-Stokes system $$ \left\{ \begin{array}{l} n_t+u\cdot\nabla n=\Delta n^m-\nabla\cdot(n\nabla c),\quad x\in \Omega, t>0, c_t+u\cdot\nabla c=\Delta c-c+n,\quad x\in \Omega, t>0,\\ u_t+\nabla P=\Delta u+n\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u=0,\quad x\in \Omega, t>0 \end{array}\right.\eqno(KSF) $$ under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$ in a three-dimensional bounded domains $\Omega\subseteq \mathbb{R}^3$ with smooth boundary, where %$\kappa\in \mathbb{R}$ is given %constant, $\phi\in W^{1,\infty}(\Omega),m>0$. %where $\phi\in W^{1,\infty}(\Omega),a \geq 0$ and $b > 0$. Here $g \in C^1(\bar{\Omega}\times[0,\infty))\cap L^\infty(\Omega\times(0,\infty))$, %$|S(x,n,c)|\leq(1+n)^{-\alpha}$ and the parameter $\alpha\geq0$. %For any small $\mu>0$, It is proved that if $m>\frac{4}{3}$, %$\alpha\geq\frac{5}{4}$ or %$m>\max\{\fra 查看全文>>