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} ρ_t+u\cdot\nablaρ=Δρ-\nabla\cdot(ρ\mathcal{S}(x,ρ,c)\nabla c)-ρm, &\!\! (x,t)\in Ω\times (0,T), \\ m_t+u\cdot\nabla m=Δm-ρm, &\!\! (x,t)\in Ω\times (0,T), \\ c_t+u\cdot\nabla c=Δc-c+m, & \!\! (x,t)\in Ω\times (0,T), \\ u_t+ (u\cdot \nabla) u=Δu-\nabla P+(ρ+m)\nablaϕ,\quad \nabla\cdot u=0, &\!\! (x,t)\in Ω\times (0,T) \end{array}\right. \end{equation*} subject to the boundary condition $(\nablaρ-ρ\mathcal{S}(x,ρ,c)\nabla c)\cdot ν\!\!=\!\nabla m\cdot ν=\nabla c\cdot ν=0, u=0$ in a bounded smooth domain $Ω\subset\mathbb R^3$. It is shown that the corresponding problem admits a globally classical solution with exponential decay properties under the hypothesis that $\mathcal{S}\in C^2(\overlineΩ\times [0,\infty)^2)^{3\times 3}$ satisfies $|\mathcal{S}(x,ρ,c)|\leq C_S$ for some $C_S>0$, and the initial data satisfy certain smallness conditions.

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