
THE SINGULAR CONVERGENCE OF A CHEMOTAXISFLUID SYSTEM MODELING CORAL FERTILIZATION^{*}
Minghua Yang, Jinyi Sun, Zunwei Fu, Zheng Wang
Acta mathematica scientia,Series B
2023, 43 (2):
492504.
DOI: 10.1007/s1047302302028
The singular convergence of a chemotaxisfluid system modeling coral fertilization is justified in spatial dimension three. More precisely, it is shown that a solution of parabolicparabolic type chemotaxisfluid 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 parabolicelliptic type chemotaxisfluid 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 FourierHerz space as $\epsilon^{1}\rightarrow 0$.
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