Global classical small-data solutions for a three-dimensional Keller--Segel--Navier--Stokes system modeling coral fertilization

We are concerned with the Keller--Segel--Navier--Stokes system \begin{equation*} \left\{ \begin{array}{ll} \rho_t+u\cdot\nabla\rho=\Delta\rho-\nabla\cdot(\rho \mathcal{S}(x,\rho,c)\nabla c)-\rho m, &\!\! (x,t)\in \Omega\times (0,T), \\ m_t+u\cdot\nabla m=\Delta m-\rho m, &\!\! (x,t)\in \Omega\times (0,T), \\ c_t+u\cdot\nabla c=\Delta c-c+m, & \!\! (x,t)\in \Omega\times (0,T), \\ u_t+ (u\cdot \nabla) u=\Delta u-\nabla P+(\rho+m)\nabla\phi,\quad \nabla\cdot u=0, &\!\! (x,t)\in \Omega\times (0,T) \end{array}\right. \end{equation*} subject to the boundary condition $(\nabla\rho-\rho \mathcal{S}(x,\rho,c)\nabla c)\cdot \nu\!\!=\!\nabla m\cdot \nu=\nabla c\cdot \nu=0, u=0$ in a bounded smooth domain $\Omega\subset\mathbb R^3$. It is shown that the corresponding problem admits a globally classical solution with exponential decay properties under the hypothesis that $\mathcal{S}\in C^2(\overline\Omega\times [0,\infty)^2)^{3\times 3}$ satisfies $|\mathcal{S}(x,\rho,c)|\leq C_S$ for some $C_S>0$, and the initial data satisfy certain smallness conditions.