Global boundedness in a two-dimensional chemotaxis-Navier-Stokes system modeling coral fertilization

We study the chemotaxis-Navier-Stokes system modeling coral fertilization(⋆)$\{\begin{array}{cc}& n_t+u\cdot \nabla n=\Delta n-\nabla \cdot \left(nS\right(x,n,c\left)\nabla c\right)-nm,\\ & c_t+u\cdot \nabla c=\Delta c-c+m,\\ & m_t+u\cdot \nabla m=\Delta m-nm,\\ & u_t+\kappa (u\cdot \nabla )u+\nabla P=\Delta u+(n+m)\nabla \varphi ,\nabla \cdot u=0\end{array}$ in a bounded and smooth domain $\Omega \subset R^2$, where $\kappa \ne 0$, $\varphi \in W^{2,\infty }\left(\Omega \right)$ and $S\in C^2(\overline{\Omega }\times {[0,\infty )}^2;R^{2\times 2})$ fulfills $\left|S\right(x,n,c\left)\right|\le S_0\left(c\right){(1+n)}^{-\alpha }$ for all $(x,n,c)\in \overline{\Omega }\times {[0,\infty )}^2$ with $\alpha \in R$ and $S_0:[0,\infty )\to [0,\infty )$ nondecreasing. In the previous work W. Wang et al. (2021) [22], we have proved that if $n\left|S\right|$ bears a superlinear growth of n with $-\frac{1}{2}<\alpha <0$, then the corresponding initial-boundary value problem of (⋆) possesses a global but not necessarily bounded solution. In the present paper, we further confirm that such a global solution must be globally bounded.