THE SINGULAR CONVERGENCE OF A CHEMOTAXIS-FLUID SYSTEM MODELING CORAL FERTILIZATION*

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THE SINGULAR CONVERGENCE OF A CHEMOTAXIS-FLUID SYSTEM MODELING CORAL FERTILIZATION^*

THE SINGULAR CONVERGENCE OF A CHEMOTAXIS-FLUID SYSTEM MODELING CORAL FERTILIZATION^*

THE SINGULAR CONVERGENCE OF A CHEMOTAXIS-FLUID SYSTEM MODELING CORAL FERTILIZATION^*

THE SINGULAR CONVERGENCE OF A CHEMOTAXIS-FLUID SYSTEM MODELING CORAL FERTILIZATION^*

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doi:10.1007/s10473-023-0202-8

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (2): 492-504.doi: 10.1007/s10473-023-0202-8

Minghua Yang¹, Jinyi Sun², Zunwei Fu^3,4,†, Zheng Wang⁵

1. Department of Mathematics, Jiangxi University of Finance and Economics, Nanchang 330032, China;
2. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China;
3. Department of Mathematics, Linyi University, Linyi 276005, China;
4. College of Information Technology, The University of Suwon, Bongdameup, Hwaseong-si, Gyeonggi-do, 445-743, Korea;
5. Department of Mathematics, The University of Suwon, Bongdameup 445-743, Korea

收稿日期:2021-07-06 修回日期:2022-02-15 出版日期:2023-03-25 发布日期:2023-04-12
通讯作者: †Zunwei Fu, E-mail: fuzunwei@lyu.edu.cn
作者简介:Minghua Yang, E-mail: minghuayang@jxufe.edu.cn; Jinyi Sun, E-mail: sunjy@nwnu.edu.cn; Zheng Wang, E-mail: wangzheng@suwon.ac.kr
基金资助:
The NSFC (12161041, 12001435 and 12071197), the training program for academic and technical leaders of major disciplines in Jiangxi Province (20204BCJL23057), the Natural Science Foundation of Jiangxi Province (20212BAB201008), the Educational Commission Science Programm of Jiangxi Province (GJJ190272) and that Natural Science Foundation of Shandong Province (ZR2021MA031).

Minghua Yang¹, Jinyi Sun², Zunwei Fu^3,4,†, Zheng Wang⁵

1. Department of Mathematics, Jiangxi University of Finance and Economics, Nanchang 330032, China;
2. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China;
3. Department of Mathematics, Linyi University, Linyi 276005, China;
4. College of Information Technology, The University of Suwon, Bongdameup, Hwaseong-si, Gyeonggi-do, 445-743, Korea;
5. Department of Mathematics, The University of Suwon, Bongdameup 445-743, Korea

Received:2021-07-06 Revised:2022-02-15 Online:2023-03-25 Published:2023-04-12
Contact: †Zunwei Fu, E-mail: fuzunwei@lyu.edu.cn
About author:Minghua Yang, E-mail: minghuayang@jxufe.edu.cn; Jinyi Sun, E-mail: sunjy@nwnu.edu.cn; Zheng Wang, E-mail: wangzheng@suwon.ac.kr
Supported by:
The NSFC (12161041, 12001435 and 12071197), the training program for academic and technical leaders of major disciplines in Jiangxi Province (20204BCJL23057), the Natural Science Foundation of Jiangxi Province (20212BAB201008), the Educational Commission Science Programm of Jiangxi Province (GJJ190272) and that Natural Science Foundation of Shandong Province (ZR2021MA031).

摘要： 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 $\epsilon^{-1}\rightarrow 0$ .

关键词: chemotaxis, singular convergence, recation, diffusion, Fourier-Herz space

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 $\begin{eqnarray*} \left\{ \begin{array}{ll} u_t^{\epsilon}+(u^{\epsilon}\cdot\nabla)u^{\epsilon}-\Delta u^{\epsilon}+\nabla\mathbf{P}^{\epsilon}=-(s^{\epsilon}+e^{\epsilon})\nabla \phi,\\ \nabla\cdot u^{\epsilon}=0, \\ e_t^{\epsilon}+(u^{\epsilon}\cdot\nabla )e^{\epsilon}-\Delta e^{\epsilon}=-s^{\epsilon}e^{\epsilon},\\ s_t^{\epsilon}+(u^{\epsilon}\cdot\nabla )s^{\epsilon}-\Delta s^{\epsilon}=-\nabla\cdot(s^{\epsilon}\nabla c^{\epsilon})-s^{\epsilon}e^{\epsilon}, \\ \epsilon^{-1} \left(c_t^{\epsilon}+(u^{\epsilon}\cdot\nabla )c^{\epsilon}\right)=\Delta c^{\epsilon}+e^{\epsilon},\\ (u^{\epsilon}, e^{\epsilon},s^{\epsilon},c^{\epsilon})|_{t=0}= (u_{0}, e_{0},s_{0},c_{0})\\ \end{array} \right. \end{eqnarray*}$ 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 $\epsilon^{-1}\rightarrow 0$ .

Key words: chemotaxis, singular convergence, recation, diffusion, Fourier-Herz space

Minghua Yang, Jinyi Sun, Zunwei Fu, Zheng Wang. THE SINGULAR CONVERGENCE OF A CHEMOTAXIS-FLUID SYSTEM MODELING CORAL FERTILIZATION^*[J]. 数学物理学报(英文版), 2023, 43(2): 492-504.

Minghua Yang, Jinyi Sun, Zunwei Fu, Zheng Wang. THE SINGULAR CONVERGENCE OF A CHEMOTAXIS-FLUID SYSTEM MODELING CORAL FERTILIZATION^*[J]. Acta mathematica scientia,Series B, 2023, 43(2): 492-504.

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