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 $ \quad n_t+u\cdot\nabla n=\Delta n^m-\nabla\cdot(n\chi(c)\nabla c), $ $ \quad c_t+u\cdot\nabla c=\Delta c-nf(c), $ $ \quad u_t+\kappa(u\cdot\nabla)u=\Delta u+\nabla P+n\nabla\Phi, $ $ \quad \nabla\cdot u=0, $ in a bounded convex domain $\Omega\subset\mathbb{R}^3$. It is proved that if $m\geq\frac{2}{3}$, $\kappa\in\mathbb{R}$, $0<\chi\in C^2([0,\infty))$, $0\leq f\in C^1([0,\infty))$ 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.