Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization

Abstract

We consider an initial-boundary value problem for the incompressible four-component Keller-Segel-Navier-Stokes system with rotational flux $$\{\begin{array}{l}{n}_t+u\cdot \nabla n=\Delta n-\nabla \cdot (nS(x,n,c)\nabla c)-nm,x\in \Omega ,t>0,\\ c_t+u\cdot \nabla c=\Delta c-c+m,x\in \Omega ,t>0,\\ m_t+u\cdot \nabla m=\Delta m-nm,x\in \Omega ,t>0,\\ u_t+\kappa (u\cdot \nabla )u+\nabla P=\Delta u+(n+m)\nabla \varphi ,x\in \Omega ,t>0,\\ \nabla \cdot u=0,x\in \Omega ,t>0\end{array}$$ in a bounded domain $\Omega \subset {\mathbb{R}}^3$ with smooth boundary, where $\kappa \in \mathbb{R}$ is given constant, $S$ is a matrix-valued sensitivity satisfying $|S(x,n,c)|\le C_S(1+n{)}^{-\alpha }$ with some $C_S>0$ and $\alpha \ge 0$. As the case $\kappa =0$ (with $\alpha \ge \frac{1}{3}$ or the initial data satisfy a certain smallness condition) has been considered in [14], based on new gradient-like functional inequality, it is shown in the present paper that the corresponding initial-boundary problem with $\kappa \ne 0$ admits at least one global weak solution if $\alpha >0$. To the best of our knowledge, this is the first analytical work for the {\bf full three-dimensional four-component} chemotaxis-Navier-Stokes system.

33. arXiv admin note: text overlap with arXiv:1806.07067