# American Institute of Mathematical Sciences

September  2020, 40(9): 5415-5439. doi: 10.3934/dcds.2020233

## Time periodic solution to a coupled chemotaxis-fluid model with porous medium diffusion

 School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

* Corresponding author: jinchhua@126.com

Received  November 2019 Revised  March 2020 Published  June 2020

Fund Project: This work is supported by NSFC(11871230), Guangdong Basic and Applied Basic Research Foundation(2020B1515310013)

This paper is concerned with the time periodic problem to a coupled chemotaxis-fluid model with porous medium diffusion $\Delta n^m$. The global existence of solutios for the initial and boundary value problem of this model have been studied by many authors, and in particular, the global solvability is established for $m>\frac65$ in dimension 3. Here, taking advantage of a double-level approximation scheme, we establish the existence of uniformly bounded time periodic solution for any $m\ge \frac 65$ and any large periodic source $g(x, t)$. In particular, the energy estimates techniques we used also applicable to the proof of global existence of the initial-boundary value problem, and one can supply the existence of global solutions for $m = \frac65$ by this method.

Citation: Jiapeng Huang, Chunhua Jin. Time periodic solution to a coupled chemotaxis-fluid model with porous medium diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (9) : 5415-5439. doi: 10.3934/dcds.2020233
##### References:

show all references

##### References:
 [1] Zhonghua Qiao, Xuguang Yang. A multiple-relaxation-time lattice Boltzmann method with Beam-Warming scheme for a coupled chemotaxis-fluid model. Electronic Research Archive, 2020, 28 (3) : 1207-1225. doi: 10.3934/era.2020066 [2] Feng Li, Yuxiang Li. Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5409-5436. doi: 10.3934/dcdsb.2019064 [3] 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 [4] Chunhua Jin. Global classical solution and stability to a coupled chemotaxis-fluid model with logistic source. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3547-3566. doi: 10.3934/dcds.2018150 [5] Guoqiang Ren, Bin Liu. Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3843-3883. doi: 10.3934/cpaa.2020170 [6] M. Ángeles Rodríguez-Bellido, Diego A. Rueda-Gómez, Élder J. Villamizar-Roa. On a distributed control problem for a coupled chemotaxis-fluid model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 557-571. doi: 10.3934/dcdsb.2017208 [7] Myeongju Chae, Kyungkeun Kang, Jihoon Lee. Existence of smooth solutions to coupled chemotaxis-fluid equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2271-2297. doi: 10.3934/dcds.2013.33.2271 [8] Jie Zhao. Large time behavior of solution to quasilinear chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1737-1755. doi: 10.3934/dcds.2020091 [9] Marco Di Francesco, Alexander Lorz, Peter A. Markowich. Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1437-1453. doi: 10.3934/dcds.2010.28.1437 [10] Jishan Fan, Kun Zhao. Improved extensibility criteria and global well-posedness of a coupled chemotaxis-fluid model on bounded domains. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3949-3967. doi: 10.3934/dcdsb.2018119 [11] Jean-Jérôme Casanova. Existence of time-periodic strong solutions to a fluid–structure system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3291-3313. doi: 10.3934/dcds.2019136 [12] Juanjuan Huang, Yan Zhou, Xuerong Shi, Zuolei Wang. A single finite-time synchronization scheme of time-delay chaotic system with external periodic disturbance. Mathematical Foundations of Computing, 2019, 2 (4) : 333-346. doi: 10.3934/mfc.2019021 [13] Changchun Liu, Pingping Li. Time periodic solutions for a two-species chemotaxis-Navier-Stokes system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020303 [14] Alexandre Vidal. Periodic orbits of tritrophic slow-fast system and double homoclinic bifurcations. Conference Publications, 2007, 2007 (Special) : 1021-1030. doi: 10.3934/proc.2007.2007.1021 [15] Xi Wang, Zuhan Liu, Ling Zhou. Asymptotic decay for the classical solution of the chemotaxis system with fractional Laplacian in high dimensions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 4003-4020. doi: 10.3934/dcdsb.2018121 [16] Lingbing He. On the global smooth solution to 2-D fluid/particle system. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 237-263. doi: 10.3934/dcds.2010.27.237 [17] Chichia Chiu, Jui-Ling Yu. An optimal adaptive time-stepping scheme for solving reaction-diffusion-chemotaxis systems. Mathematical Biosciences & Engineering, 2007, 4 (2) : 187-203. doi: 10.3934/mbe.2007.4.187 [18] Hiroko Morimoto. Survey on time periodic problem for fluid flow under inhomogeneous boundary condition. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 631-639. doi: 10.3934/dcdss.2012.5.631 [19] Masahiro Kubo, Noriaki Yamazaki. Periodic stability of elliptic-parabolic variational inequalities with time-dependent boundary double obstacles. Conference Publications, 2007, 2007 (Special) : 614-623. doi: 10.3934/proc.2007.2007.614 [20] Norikazu Saito. Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis. Communications on Pure & Applied Analysis, 2012, 11 (1) : 339-364. doi: 10.3934/cpaa.2012.11.339

2019 Impact Factor: 1.338