Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion

Abstract

We consider an initial-boundary value problem for the incompressible chemotaxis-Navier-Stokes equations generalizing the porous-medium-type diffusion model $n_t+u\cdot \nabla n=\Delta n^m-\nabla \cdot (n\chi (c)\nabla c),$ $c_t+u\cdot \nabla c=\Delta c-nf(c),$ $u_t+\kappa (u\cdot \nabla )u=\Delta u+\nabla P+n\nabla \Phi ,$ $\nabla \cdot u=0,$ in a bounded convex domain $\Omega \subset {\mathbb{R}}^3$. It is proved that if $m\ge \frac{2}{3}$, $\kappa \in \mathbb{R}$, $0<\chi \in C^2([0,\infty ))$, $0\le f\in C^1([0,\infty ))$ with $f(0)=0$ and $\Phi \in W^{1,\infty }(\Omega )$, then for sufficiently smooth initial data $(n_0,c_0,u_0)$ the model possesses at least one global weak solution.