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+ucdot ablarho=Deltarho- ablacdot(rho mathcal{S}(x,rho,c) abla c)-rho m, &!! (x,t)in Omegatimes (0,T), m_t+ucdot abla m=Delta m-rho m, &!! (x,t)in Omegatimes (0,T), c_t+ucdot abla c=Delta c-c+m, & !! (x,t)in Omegatimes (0,T), u_t+ (ucdot abla) u=Delta u- abla P+(rho+m) ablaphi,quad ablacdot u=0, &!! (x,t)in Omegatimes (0,T) end{array}right. end{equation*} subject to the boundary condition $( ablarho-rho mathcal{S}(x,rho,c) abla c)cdot u!!=! abla mcdot u= abla ccdot u=0, u=0$ in a bounded smooth domain $Omegasubsetmathbb R^3$. It is shown that the corresponding problem admits a globally classical solution with exponential decay properties under the hypothesis that $mathcal{S}in C^2(overlineOmegatimes [0,infty)^2)^{3times 3}$ satisfies $|mathcal{S}(x,rho,c)|leq C_S $ for some $C_S>0$, and the initial data satisfy certain smallness conditions.