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Boundedness of weak solutions in a 3D chemotaxis-Stokes system with nonlinear double degeneracy diffusion
  • Qingyun Lin
Qingyun Lin
University of Electronic Science and Technology of China

Corresponding Author:[email protected]

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Abstract

In this paper, an incompressible chemotaxis-Stokes system with nonlinear double degeneracy diffusion \begin{equation*} \left\{ \begin{split} &n_t+{\bf u}\cdot\nabla n=\nabla\cdot({|\nabla n^m|}^{p-2} \nabla n^m)-\nabla\cdot\big(n\nabla c\big),& \qquad x\in\Omega,\,t>0,\\ &c_t+{\bf u}\cdot\nabla c=\Delta c-nc, &\qquad x\in\Omega,\,t>0,\\ &{\bf u}_t=\Delta{\bf u}+\nabla P+n\nabla\phi,&\qquad x\in\Omega,\,t>0,\\ &\nabla\cdot{\bf u}=0,&\qquad x\in\Omega,\,t>0\\ \end{split} \right. \end{equation*} \\ is considered in a smooth bounded domain $\Omega\subset {\mathbb{R}}^3$, where potential function $\phi\in W^{2,\infty}(\Omega)$ is given. Here we have homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for ${\bf u}$. It is proved that global bounded weak solutions exist whenever $8mp-8m+3p>15$, $p\geq2$ and $m\geq1$.