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THE SINGULAR CONVERGENCE OF A CHEMOTAXIS-FLUID SYSTEM MODELING CORAL FERTILIZATION*
Minghua Yang, Jinyi Sun, Zunwei Fu, Zheng Wang
Acta mathematica scientia,Series B. 2023, 43 (2):
492-504.
DOI: 10.1007/s10473-023-0202-8
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 $\begin{eqnarray*} \left\{ \begin{array}{ll} u_t^{\epsilon}+(u^{\epsilon}\cdot\nabla)u^{\epsilon}-\Delta u^{\epsilon}+\nabla\mathbf{P}^{\epsilon}=-(s^{\epsilon}+e^{\epsilon})\nabla \phi,\\ \nabla\cdot u^{\epsilon}=0, \\ e_t^{\epsilon}+(u^{\epsilon}\cdot\nabla )e^{\epsilon}-\Delta e^{\epsilon}=-s^{\epsilon}e^{\epsilon},\\ s_t^{\epsilon}+(u^{\epsilon}\cdot\nabla )s^{\epsilon}-\Delta s^{\epsilon}=-\nabla\cdot(s^{\epsilon}\nabla c^{\epsilon})-s^{\epsilon}e^{\epsilon}, \\ \epsilon^{-1} \left(c_t^{\epsilon}+(u^{\epsilon}\cdot\nabla )c^{\epsilon}\right)=\Delta c^{\epsilon}+e^{\epsilon},\\ (u^{\epsilon}, e^{\epsilon},s^{\epsilon},c^{\epsilon})|_{t=0}= (u_{0}, e_{0},s_{0},c_{0})\\ \end{array} \right. \end{eqnarray*}$ converges to that of the parabolic-elliptic type chemotaxis-fluid system modeling coral fertilization $\begin{eqnarray*} \left\{ \begin{array}{ll} u_t^{\infty}+(u^{\infty}\cdot\nabla)u^{\infty}-\Delta u^{\infty}+\nabla\mathbf{P}^{\infty}=-(s^{\infty}+e^{\infty})\nabla \phi, \\ \nabla\cdot u^{\infty}=0, \\ e_t^{\infty}+(u^{\infty}\cdot\nabla )e^{\infty}-\Delta e^{\infty}=-s^{\infty}e^{\infty}, \\ s_t^{\infty}+(u^{\infty}\cdot\nabla )s^{\infty}-\Delta s^{\infty}=-\nabla\cdot(s^{\infty}\nabla c^{\infty})-s^{\infty}e^{\infty}, \\ 0=\Delta c^{\infty}+e^{\infty}, \\ (u^{\infty}, e^{\infty},s^{\infty})|_{t=0}= (u_{0}, e_{0},s_{0})\\ \end{array} \right. \end{eqnarray*}$ in a certain Fourier-Herz space as $\epsilon^{-1}\rightarrow 0$.
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