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

We consider an initial-boundary value problem for the incompressible four-component Keller-Segel-Navier-Stokes system with rotational flux

$ \begin{align} \left\{ \begin{array}{l} n_t+u\cdot\nabla n = \Delta n-\nabla\cdot(nS(x,n,c)\nabla c)-nm,\quad x\in \Omega, t>0,\\ c_t+u\cdot\nabla c = \Delta c-c+m,\quad x\in \Omega, t>0,\\ m_t+u\cdot\nabla m = \Delta m-nm,\quad x\in \Omega, t>0, \qquad \qquad \qquad \qquad \qquad \qquad (*)\\ u_t+\kappa(u \cdot \nabla)u+\nabla P = \Delta u+(n+m)\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u = 0,\quad x\in \Omega, t>0\\ \end{array}\right. \end{align} $

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)|\leq C_S(1+n)^{-\alpha} $ with some $ C_S> 0 $ and $ \alpha\geq 0 $. As the case $ \kappa = 0 $ (with $ \alpha\geq\frac{1}{3} $ or the initial data satisfy a certain smallness condition) has been considered in [18], based on new gradient-like functional inequality, it is shown in the present paper that the corresponding initial-boundary problem with $ \kappa \neq 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 full three-dimensional four-component chemotaxis-Navier-Stokes system.