Global weak solutions to a 3-dimensional degenerate and singular chemotaxis-Navier--Stokes system with logistic source

Abstract

This paper considers the degenerate and singular chemotaxis-Navier--Stokes system with logistic term $n_t+u\cdot \nabla n=\Delta n^m-\chi \nabla \cdot (n\nabla c)+\kappa n-\mu n^2$, $x\in \Omega ,t>0$, $c_t+u\cdot \nabla c=\Delta c-nc$, $x\in \Omega ,t>0$, $u_t+(u\cdot \nabla )u=\Delta u+\nabla P+n\nabla \Phi ,\nabla \cdot u=0$, $x\in \Omega ,t>0$, where $\Omega \subset {\mathbb{R}}^3$ is a bounded domain and $\chi ,\kappa \ge 0$ and $m,\mu >0$. In the above system without fluid environment Jin (J. Differential Equations, 2017) showed existence and boundedness of global weak solutions. On the other hand, in the above system with $m=1$, Lankeit (Math.\ Models Methods Appl. Sci., 2016) established global existence of weak solutions. However, the above system with $m>0$ has not been studied yet. The purpose of this talk is to establish global existence of weak solutions in the chemotaxis-Navier--Stokes system with degenerate diffusion and logistic term.

21 pages