Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion

Abstract

The coupled quasilinear Keller-Segel-Navier-Stokes system is considered under Neumann boundary conditions for $n$ and $c$ and no-slip boundary conditions for $u$ in three-dimensional bounded domains $\Omega \subseteq {\mathbb{R}}^3$ with smooth boundary, where $m>0,\kappa \in \mathbb{R}$ are given constants, $\varphi \in W^{1,\infty }(\Omega )$. If $m>2$, then for all reasonably regular initial data, a corresponding initial-boundary value problem for $(KSNF)$ possesses a globally defined weak solution.