| Abstract: |
| In this talk we present an incompressible chemotaxis-Navier-Stokes system with slow $p$-Laplacian diffusion
\begin{eqnarray}\nonumber
\left\{\begin{array}{lll}
\medskip
n_t+u\cdot\nabla n=\nabla\cdot(|\nabla n|^{p-2}\nabla n)-\nabla\cdot(n\chi(c)\nabla c),&{} x\in\Omega,\ t>0,\
\medskip
c_t+u\cdot\nabla c=\Delta c-nf(c),&{} x\in\Omega,\ t>0,\
\medskip
u_t+\kappa(u\cdot\nabla) u=\Delta u+\nabla P+n\nabla\Phi,&{} x\in\Omega,\ t>0,\
\medskip
\nabla\cdot u=0,&{} x\in\Omega,\ t>0
\end{array}\right.
\end{eqnarray}
under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$ in a bounded convex domain $\Omega\subset \mathbb{R}^3$ with smooth boundary. Here, $\Phi\in W^{1,\infty}(\Omega)$, $0\frac{32}{15}$ and under appropriate structural assumptions on $f$ and $\chi$, for all sufficiently smooth initial data $(n_0,c_0,u_0)$ the model possesses at least one global weak solution. |
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