Global weak solutions to a 3D self-consistent chemotaxis-Stokes system with nonlinear resource consumption

Abstract

summary:We study the self-consistent chemotaxis-fluid system with nonlinear resource consumption $$\begin {cases} n_{t}+u\cdot \nabla n=\Delta n^m -\nabla \cdot (n \nabla c)+\nabla \cdot (n\nabla \phi ), & x\in \Omega ,\ t>0, \\ c_{t}+u\cdot \nabla c=\Delta c-n^\alpha c, & x\in \Omega ,\ t>0, \\ u_t+ \nabla P=\Delta u-n\nabla \phi +n \nabla c,& x\in \Omega ,\ t>0,\\ \nabla \cdot u=0,& x\in \Omega ,\ t>0,\\ \end {cases}$$ under no-flux boundary conditions in a bounded domain $\Omega \subset \mathbb {R}^3$ with smooth boundary. It is proved that this system possesses a global weak solution provided $m>1$ and $\alpha > \frac {4}{3}$