Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with slow $p$-Laplacian diffusion

Abstract

This paper investigates an incompressible chemotaxis-Navier-Stokes system with slow $p$-Laplacian diffusion \begin{eqnarray} \left\{\begin{array}{lll} n_t+u\cdot\nabla n=\nabla\cdot(|\nabla n|^{p-2}\nabla n)-\nabla\cdot(nχ(c)\nabla c),& x\inΩ,\ t>0, c_t+u\cdot\nabla c=Δc-nf(c),& x\inΩ,\ t>0, u_t+(u\cdot\nabla) u=Δu+\nabla P+n\nablaΦ,& x\inΩ,\ t>0, \nabla\cdot u=0,& x\inΩ,\ 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, $\Phi \in W^{1,\infty }(\Omega )$, $0<\chi \in C^2([0,\infty ))$ and $0\le f\in C^1([0,\infty ))$ with $f(0)=0$. It is proved that if $p>\frac{32}{15}$ 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.

22pages. arXiv admin note: text overlap with arXiv:1501.05171