Asymptotic behavior of classical solutions of a three-dimensional Keller–Segel–Navier–Stokes system modeling coral fertilization

Abstract

We are concerned with the Keller–Segel–Navier–Stokes system

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \rho _t+u\cdot \nabla \rho =\Delta \rho -\nabla \cdot (\rho \mathcal {S}(x,\rho ,c)\nabla c)-\rho m, &{}\quad (x,t)\in \varOmega \times (0,T),\\ m_t+u\cdot \nabla m=\Delta m-\rho m, &{}\quad (x,t)\in \varOmega \times (0,T),\\ c_t+u\cdot \nabla c=\Delta c-c+m, &{}\quad (x,t)\in \varOmega \times (0,T),\\ u_t+ (u\cdot \nabla ) u=\Delta u-\nabla P+(\rho +m)\nabla \phi ,\quad \nabla \cdot u=0, &{}\quad (x,t)\in \varOmega \times (0,T) \end{array}\right. \end{aligned}$$

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 $\varOmega \subset \mathbb {R}^3$. It is shown that this problem admits a global classical solution with exponential decay properties when $\mathcal {S}\in C^2(\overline{\varOmega }\times [0,\infty )^2)^{3\times 3}$ satisfies $|\mathcal {S}(x,\rho ,c)|\le C_S$ for some $C_S>0$, and the initial data satisfy certain smallness conditions.