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April  2020, 40(4): 2135-2163. doi: 10.3934/dcds.2020109

Global well-posedness and long time behaviors of chemotaxis-fluid system modeling coral fertilization

1. 

Department of Applied Mathematics and Institute for integrated Mathematical Sciences, Hankyong National University, Ansung, Korea

2. 

Department of Mathematics, Yonsei University, Seoul, Korea

3. 

Department of Mathematics, Chung-Ang University, Seoul, Korea

* Corresponding author: Jihoon Lee

Received  April 2019 Revised  October 2019 Published  January 2020

We consider generalized models on coral broadcast spawning phenomena involving diffusion, advection, chemotaxis, and reactions when egg and sperm densities are different. We prove the global-in-time existence of the regular solutions of the models as well as their temporal decays in two and three dimensions. We also show that the total masses of egg and sperm density have positive lower bounds as time tends to infinity in three dimensions.

Citation: Myeongju Chae, Kyungkeun Kang, Jihoon Lee. Global well-posedness and long time behaviors of chemotaxis-fluid system modeling coral fertilization. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2135-2163. doi: 10.3934/dcds.2020109
References:
[1]

J. AhnK. KangJ. Kim and J. Lee, Lower bound of mass in a chemotactic model with advection and absorbing reaction, SIAM J. Math. Anal., 49 (2017), 723-755.  doi: 10.1137/16M1071778.  Google Scholar

[2]

M. ChaeK. Kang and J. Lee, Asymptotic behaviors of solutions for an aerotaxis model coupled to fluid equations, J. Korean Math. Soc., 53 (2016), 127-146.  doi: 10.4134/JKMS.2016.53.1.127.  Google Scholar

[3]

M. ChaeK. Kang and J. Lee, Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Partial Differential Equations, 39 (2014), 1205-1235.  doi: 10.1080/03605302.2013.852224.  Google Scholar

[4]

M. ChaeK. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297.  doi: 10.3934/dcds.2013.33.2271.  Google Scholar

[5]

M. ChaeK. KangJ. Lee and K.-A. Lee, A regularity condition and temporal asymptotics for chemotaxis-fluid equations, Nonlinearity, 31 (2018), 351-387.  doi: 10.1088/1361-6544/aa92ec.  Google Scholar

[6]

J. C. Coll and et al., Chemical aspects of mass spawning in corals. I. Sperm-attractant molecules in the eggs of the scleractinian coral Montipora digitata, Mar. Biol., 118 (1994), 177-182. Google Scholar

[7]

J. C. Coll and et al., Chemical aspects of mass spawning in corals. Ⅱ. Epi-thunbergol, the sperm attractant in the egg of the soft coral Lobophytum crassum (Cnidaria: Octocorallia), Mar. Biol., 123 (1995), 137-143. Google Scholar

[8]

M. W. Denny and M. F. Shibata, Consequences of surf-sone turbulence for settlement and external fertilization, Am. Nat, 134 (1989), 859-889.   Google Scholar

[9]

J. E. Eckman, Closing the larval loop: Linking larval ecology to the population dynamics of marine benthic invertebrates, J. Exp. Mar. Biol. Ecol., 200 (1996), 207-237.  doi: 10.1016/S0022-0981(96)02644-5.  Google Scholar

[10]

E. Espejo and T. Suzuki, Reaction terms avoiding aggregation in slow fluids, Nonlinear Anal. Real World Appl., 21 (2015), 110-126.  doi: 10.1016/j.nonrwa.2014.07.001.  Google Scholar

[11]

E. Espejo and M. Winkler, Global classical solvability and stabilization in a two-dimensional chemotaxis-Navier-Stokes system modeling coral fertilization, Nonlinearity, 31 (2018), 1227-1259.  doi: 10.1088/1361-6544/aa9d5f.  Google Scholar

[12]

Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications for Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94.  doi: 10.1016/0022-1236(91)90136-S.  Google Scholar

[13]

E. F. Keller and L. A. Segel, Initiation of slide mold aggregation viewd as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[14]

E. F. Keller and L.A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[15]

A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and enhancement of biological reactions, Comm. Partial Differential Equations, 37 (2012), 298-318.  doi: 10.1080/03605302.2011.589879.  Google Scholar

[16]

A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and efficiency of biological reactions: The critical reaction case, J. Math. Phys., 53 (2012), 115609, 9 pp. doi: 10.1063/1.4742858.  Google Scholar

[17]

H. Lasker, High fertilization success in a surface brooding Carribean Gorgonian, Biol. Bull., 210 (2006), 10-17.   Google Scholar

[18]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biol. Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407.  Google Scholar

[19]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Commun. PDE., 32 (2007), 849-977.  doi: 10.1080/03605300701319003.  Google Scholar

[20]

M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.  doi: 10.1016/j.jmaa.2008.07.071.  Google Scholar

[21]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logisitic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.  Google Scholar

show all references

References:
[1]

J. AhnK. KangJ. Kim and J. Lee, Lower bound of mass in a chemotactic model with advection and absorbing reaction, SIAM J. Math. Anal., 49 (2017), 723-755.  doi: 10.1137/16M1071778.  Google Scholar

[2]

M. ChaeK. Kang and J. Lee, Asymptotic behaviors of solutions for an aerotaxis model coupled to fluid equations, J. Korean Math. Soc., 53 (2016), 127-146.  doi: 10.4134/JKMS.2016.53.1.127.  Google Scholar

[3]

M. ChaeK. Kang and J. Lee, Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Partial Differential Equations, 39 (2014), 1205-1235.  doi: 10.1080/03605302.2013.852224.  Google Scholar

[4]

M. ChaeK. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297.  doi: 10.3934/dcds.2013.33.2271.  Google Scholar

[5]

M. ChaeK. KangJ. Lee and K.-A. Lee, A regularity condition and temporal asymptotics for chemotaxis-fluid equations, Nonlinearity, 31 (2018), 351-387.  doi: 10.1088/1361-6544/aa92ec.  Google Scholar

[6]

J. C. Coll and et al., Chemical aspects of mass spawning in corals. I. Sperm-attractant molecules in the eggs of the scleractinian coral Montipora digitata, Mar. Biol., 118 (1994), 177-182. Google Scholar

[7]

J. C. Coll and et al., Chemical aspects of mass spawning in corals. Ⅱ. Epi-thunbergol, the sperm attractant in the egg of the soft coral Lobophytum crassum (Cnidaria: Octocorallia), Mar. Biol., 123 (1995), 137-143. Google Scholar

[8]

M. W. Denny and M. F. Shibata, Consequences of surf-sone turbulence for settlement and external fertilization, Am. Nat, 134 (1989), 859-889.   Google Scholar

[9]

J. E. Eckman, Closing the larval loop: Linking larval ecology to the population dynamics of marine benthic invertebrates, J. Exp. Mar. Biol. Ecol., 200 (1996), 207-237.  doi: 10.1016/S0022-0981(96)02644-5.  Google Scholar

[10]

E. Espejo and T. Suzuki, Reaction terms avoiding aggregation in slow fluids, Nonlinear Anal. Real World Appl., 21 (2015), 110-126.  doi: 10.1016/j.nonrwa.2014.07.001.  Google Scholar

[11]

E. Espejo and M. Winkler, Global classical solvability and stabilization in a two-dimensional chemotaxis-Navier-Stokes system modeling coral fertilization, Nonlinearity, 31 (2018), 1227-1259.  doi: 10.1088/1361-6544/aa9d5f.  Google Scholar

[12]

Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications for Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94.  doi: 10.1016/0022-1236(91)90136-S.  Google Scholar

[13]

E. F. Keller and L. A. Segel, Initiation of slide mold aggregation viewd as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[14]

E. F. Keller and L.A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[15]

A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and enhancement of biological reactions, Comm. Partial Differential Equations, 37 (2012), 298-318.  doi: 10.1080/03605302.2011.589879.  Google Scholar

[16]

A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and efficiency of biological reactions: The critical reaction case, J. Math. Phys., 53 (2012), 115609, 9 pp. doi: 10.1063/1.4742858.  Google Scholar

[17]

H. Lasker, High fertilization success in a surface brooding Carribean Gorgonian, Biol. Bull., 210 (2006), 10-17.   Google Scholar

[18]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biol. Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407.  Google Scholar

[19]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Commun. PDE., 32 (2007), 849-977.  doi: 10.1080/03605300701319003.  Google Scholar

[20]

M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.  doi: 10.1016/j.jmaa.2008.07.071.  Google Scholar

[21]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logisitic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.  Google Scholar

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