The Singular Convergence of a Chemotaxis-Fluid System Modeling Coral Fertilization

Abstract

The singular convergence of a Chemotaxis-fluid system modeling coral fertilization is justified in spatial dimension three. More precisely, it is shown that a solution of parabolic-parabolic type chemotaxis-fluid system modeling coral fertilization

$$\left\{ {\begin{array}{*{20}{c}} {u_t^ \in+ ({u^ \in } \cdot \nabla ){u^ \in } - \Delta {u^ \in } + \nabla {P^ \in } =- ({s^ \in } + {e^ \in })\nabla \phi ,} \\ {\nabla\cdot {u^ \in } = 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \\ {e_t^ \in+ ({u^ \in } \cdot \nabla ){e^ \in } - \Delta {e^ \in } =- {s^ \in }{e^ \in }\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \\ {s_t^ \in+ ({u^ \in } \cdot \nabla ){s^ \in } - \Delta {s^ \in } =- \nabla\cdot ({s^ \in }\nabla {c^ \in }) - {s^ \in }{e^ \in },\;\;} \\ {{ \in ^{ - 1}}(c_t^ \in+ ({u^ \in } \cdot \nabla ){c^ \in }) = \Delta {c^ \in } + {e^ \in },\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \\ {({u^ \in },{e^ \in },{s^ \in },{c^ \in }){|_{t = 0}} = ({u_0},{e_0},{s_0},{c_0})\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \end{array}} \right.$$

converges to that of the parabolic-elliptic type chemotaxis-fluid system modeling coral fertilization

$$\left\{ {\matrix{{u_t^\infty + \left( {{u^\infty } \cdot \nabla } \right){u^\infty } - \Delta {u^\infty } + \nabla {{\bf{P}}^\infty } = - \left( {{s^\infty } + {e^\infty }} \right)\nabla \phi ,} \hfill \cr {\nabla \cdot \le {u^\infty } = 0,} \hfill \cr {e_t^\infty + \left( {{u^\infty } \cdot \nabla } \right){e^\infty } - \Delta {e^\infty } = - {s^\infty }{e^\infty },} \hfill \cr {s_t^\infty + \left( {{u^\infty } \cdot \nabla } \right){s^\infty } - \Delta {s^\infty } = - \nabla \cdot \left( {{s^\infty }\nabla {c^\infty }} \right) - {s^\infty }{e^\infty },} \hfill \cr {0 = \Delta {c^\infty } + {e^\infty },} \hfill \cr {\left( {{u^\infty },{e^\infty },{s^\infty }} \right)\left| {_{t = 0}} \right. = \left( {{u_0},{e_0},{s_0}} \right)} \hfill \cr } } \right.$$

in a certain Fourier-Herz space as ϵ⁻¹ → 0.