• Previous Article
    Multiple-site deep brain stimulation with delayed rectangular waveforms for Parkinson's disease
  • ERA Home
  • This Issue
  • Next Article
    Global stability of traveling wave fronts in a two-dimensional lattice dynamical system with global interaction
doi: 10.3934/era.2021050
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Global existence for a two-species chemotaxis-Navier-Stokes system with $ p $-Laplacian

Department of Mathematics, Jilin University, Changchun 130012, China

* Corresponding author: Changchun Liu

Received  February 2021 Revised  May 2021 Early access July 2021

Fund Project: This work is supported by the Jilin Scientific and Technological Development Program

We consider a two-species chemotaxis-Navier-Stokes system with $ p $-Laplacian in three-dimensional smooth bounded domains. It is proved that for any $ p\geq2 $, the problem admits a global weak solution.

Citation: Jiayi Han, Changchun Liu. Global existence for a two-species chemotaxis-Navier-Stokes system with $ p $-Laplacian. Electronic Research Archive, doi: 10.3934/era.2021050
References:
[1]

X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.  doi: 10.1512/iumj.2016.65.5776.  Google Scholar

[2]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.  Google Scholar

[3]

X. CaoS. Kurima and M. Mizukami, Global existence and asymptotic behavior of classical solutions for a 3D two-species chemotaxis-Stokes system with competitive kinetics, Math. Meth. Appl. Sci., 41 (2018), 3138-3154.  doi: 10.1002/mma.4807.  Google Scholar

[4]

R. Dal PassoH. Garcke and G. Grün, On a fourth-order degenerate parabolic equation: global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anal., 29 (1998), 321-342.  doi: 10.1137/S0036141096306170.  Google Scholar

[5]

K. FujieA. Ito and T. Yokota, Existence and uniqueness of local classical solutions to modified tumor invasion models of Chaplain-Anderson type, Adv. Math. Sci. Appl., 24 (2014), 67-84.   Google Scholar

[6]

M. HirataS. KurimaM. Mizukami and T. Yokota, Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, J. Differential Equations, 263 (2017), 470-490.  doi: 10.1016/j.jde.2017.02.045.  Google Scholar

[7]

C. Jin, Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1675-1688.  doi: 10.3934/dcdsb.2018069.  Google Scholar

[8]

H.-Y. Jin and T. Xiang, Convergence rates of solutions for a two-species chemotaxis-Navier-Stokes sytstem with competitive kinetics, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1919-1942.  doi: 10.3934/dcdsb.2018249.  Google Scholar

[9]

J.-L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[10]

J. Liu, Boundedness in a chemotaxis-Navier-Stokes System modeling coral fertilization with slow $p$-Laplacian diffusion, J. Math. Fluid Mech., 22 (2020), No. 10, 31 pp. doi: 10.1007/s00021-019-0469-7.  Google Scholar

[11]

C. Liu and P. Li, Global existence for a chemotaxis-haptotaxis model with $p$-Laplacian, Commun. Pure Appl. Anal., 19 (2020), 1399-1419.  doi: 10.3934/cpaa.2020070.  Google Scholar

[12]

C. Liu and P. Li, Boundedness and global solvability for a chemotaxis-haptotaxis model with $p$-Laplacian diffusion, Electron. J. Differential Equations, (2020), Paper No. 16, 16 pp.  Google Scholar

[13]

C. Liu and P. Li, Time periodic solutions for a two-species chemotaxis-Navier-Stokes system, Discrete and Continuous Dynamical Systems Series B, 26 (2021), 4567-4585.  doi: 10.3934/dcdsb.2020303.  Google Scholar

[14]

T. Miyakawa and H. Sohr, On energy inequality, smoothness and large time behaviour in $L^2$ for weak solutions of the Navier-Stokes equations in exterior domains, Math. Z., 199 (1988), 455-478.  doi: 10.1007/BF01161636.  Google Scholar

[15]

M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2301-2319.  doi: 10.3934/dcdsb.2017097.  Google Scholar

[16]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162.   Google Scholar

[17]

H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach, Birkhäuser, Basel, 2001.  Google Scholar

[18]

C. StinnerC. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.  doi: 10.1137/13094058X.  Google Scholar

[19]

W. Tao and Y. Li, Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with slow $p$-Laplacian diffusion, Nonlinear Anal. Real World Appl., 45 (2019), 26-52.  doi: 10.1016/j.nonrwa.2018.06.005.  Google Scholar

[20]

W. Tao and Y. Li, Boundedness of weak solutions of a chemotaxis-Stokes system with slow $p$-Laplacian diffusion, J. Differential Equations, 268 (2020), 6872-6919.  doi: 10.1016/j.jde.2019.11.078.  Google Scholar

[21]

I. TuvalL. CisnerosC. DombrowskiC. W. WolgemuthJ. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277-2282.  doi: 10.1073/pnas.0406724102.  Google Scholar

[22]

M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352.  doi: 10.1016/j.anihpc.2015.05.002.  Google Scholar

[23]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.  Google Scholar

[24]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[25]

M. Winkler, How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?, Trans. Amer. Math. Soc., 369 (2017), 3067-3125.  doi: 10.1090/tran/6733.  Google Scholar

show all references

References:
[1]

X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.  doi: 10.1512/iumj.2016.65.5776.  Google Scholar

[2]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.  Google Scholar

[3]

X. CaoS. Kurima and M. Mizukami, Global existence and asymptotic behavior of classical solutions for a 3D two-species chemotaxis-Stokes system with competitive kinetics, Math. Meth. Appl. Sci., 41 (2018), 3138-3154.  doi: 10.1002/mma.4807.  Google Scholar

[4]

R. Dal PassoH. Garcke and G. Grün, On a fourth-order degenerate parabolic equation: global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anal., 29 (1998), 321-342.  doi: 10.1137/S0036141096306170.  Google Scholar

[5]

K. FujieA. Ito and T. Yokota, Existence and uniqueness of local classical solutions to modified tumor invasion models of Chaplain-Anderson type, Adv. Math. Sci. Appl., 24 (2014), 67-84.   Google Scholar

[6]

M. HirataS. KurimaM. Mizukami and T. Yokota, Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, J. Differential Equations, 263 (2017), 470-490.  doi: 10.1016/j.jde.2017.02.045.  Google Scholar

[7]

C. Jin, Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1675-1688.  doi: 10.3934/dcdsb.2018069.  Google Scholar

[8]

H.-Y. Jin and T. Xiang, Convergence rates of solutions for a two-species chemotaxis-Navier-Stokes sytstem with competitive kinetics, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1919-1942.  doi: 10.3934/dcdsb.2018249.  Google Scholar

[9]

J.-L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[10]

J. Liu, Boundedness in a chemotaxis-Navier-Stokes System modeling coral fertilization with slow $p$-Laplacian diffusion, J. Math. Fluid Mech., 22 (2020), No. 10, 31 pp. doi: 10.1007/s00021-019-0469-7.  Google Scholar

[11]

C. Liu and P. Li, Global existence for a chemotaxis-haptotaxis model with $p$-Laplacian, Commun. Pure Appl. Anal., 19 (2020), 1399-1419.  doi: 10.3934/cpaa.2020070.  Google Scholar

[12]

C. Liu and P. Li, Boundedness and global solvability for a chemotaxis-haptotaxis model with $p$-Laplacian diffusion, Electron. J. Differential Equations, (2020), Paper No. 16, 16 pp.  Google Scholar

[13]

C. Liu and P. Li, Time periodic solutions for a two-species chemotaxis-Navier-Stokes system, Discrete and Continuous Dynamical Systems Series B, 26 (2021), 4567-4585.  doi: 10.3934/dcdsb.2020303.  Google Scholar

[14]

T. Miyakawa and H. Sohr, On energy inequality, smoothness and large time behaviour in $L^2$ for weak solutions of the Navier-Stokes equations in exterior domains, Math. Z., 199 (1988), 455-478.  doi: 10.1007/BF01161636.  Google Scholar

[15]

M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2301-2319.  doi: 10.3934/dcdsb.2017097.  Google Scholar

[16]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162.   Google Scholar

[17]

H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach, Birkhäuser, Basel, 2001.  Google Scholar

[18]

C. StinnerC. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.  doi: 10.1137/13094058X.  Google Scholar

[19]

W. Tao and Y. Li, Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with slow $p$-Laplacian diffusion, Nonlinear Anal. Real World Appl., 45 (2019), 26-52.  doi: 10.1016/j.nonrwa.2018.06.005.  Google Scholar

[20]

W. Tao and Y. Li, Boundedness of weak solutions of a chemotaxis-Stokes system with slow $p$-Laplacian diffusion, J. Differential Equations, 268 (2020), 6872-6919.  doi: 10.1016/j.jde.2019.11.078.  Google Scholar

[21]

I. TuvalL. CisnerosC. DombrowskiC. W. WolgemuthJ. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277-2282.  doi: 10.1073/pnas.0406724102.  Google Scholar

[22]

M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352.  doi: 10.1016/j.anihpc.2015.05.002.  Google Scholar

[23]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.  Google Scholar

[24]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[25]

M. Winkler, How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?, Trans. Amer. Math. Soc., 369 (2017), 3067-3125.  doi: 10.1090/tran/6733.  Google Scholar

[1]

Laiqing Meng, Jia Yuan, Xiaoxin Zheng. Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3413-3441. doi: 10.3934/dcds.2019141

[2]

Yulan Wang. Global solvability in a two-dimensional self-consistent chemotaxis-Navier-Stokes system. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 329-349. doi: 10.3934/dcdss.2020019

[3]

Minghua Yang, Zunwei Fu, Jinyi Sun. Global solutions to Chemotaxis-Navier-Stokes equations in critical Besov spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3427-3460. doi: 10.3934/dcdsb.2018284

[4]

Mimi Dai, Han Liu. Low modes regularity criterion for a chemotaxis-Navier-Stokes system. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2713-2735. doi: 10.3934/cpaa.2020118

[5]

Sachiko Ishida. Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3463-3482. doi: 10.3934/dcds.2015.35.3463

[6]

Xiaoping Zhai, Zhaoyang Yin. Global solutions to the Chemotaxis-Navier-Stokes equations with some large initial data. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2829-2859. doi: 10.3934/dcds.2017122

[7]

Qingshan Zhang, Yuxiang Li. Convergence rates of solutions for a two-dimensional chemotaxis-Navier-Stokes system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2751-2759. doi: 10.3934/dcdsb.2015.20.2751

[8]

Changchun Liu, Pingping Li. Time periodic solutions for a two-species chemotaxis-Navier-Stokes system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4567-4585. doi: 10.3934/dcdsb.2020303

[9]

Chengxin Du, Changchun Liu. Time periodic solution to a two-species chemotaxis-Stokes system with $ p $-Laplacian diffusion. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021162

[10]

Changchun Liu, Pingping Li. Global existence for a chemotaxis-haptotaxis model with $ p $-Laplacian. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1399-1419. doi: 10.3934/cpaa.2020070

[11]

Hai-Yang Jin, Tian Xiang. Convergence rates of solutions for a two-species chemotaxis-Navier-Stokes sytstem with competitive kinetics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1919-1942. doi: 10.3934/dcdsb.2018249

[12]

Jian-Guo Liu, Zhaoyun Zhang. Existence of global weak solutions of $ p $-Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021051

[13]

Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the existence of solutions for the Navier-Stokes system in a sum of weak-$L^{p}$ spaces. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 171-183. doi: 10.3934/dcds.2010.27.171

[14]

Frederic Heihoff. Global mass-preserving solutions for a two-dimensional chemotaxis system with rotational flux components coupled with a full Navier–Stokes equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4703-4719. doi: 10.3934/dcdsb.2020120

[15]

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

[16]

Jinyi Sun, Zunwei Fu, Yue Yin, Minghua Yang. Global existence and Gevrey regularity to the Navier-Stokes-Nernst-Planck-Poisson system in critical Besov-Morrey spaces. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3409-3425. doi: 10.3934/dcdsb.2020237

[17]

Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361

[18]

Yutaka Tsuzuki. Solvability of $p$-Laplacian parabolic logistic equations with constraints coupled with Navier-Stokes equations in 2D domains. Evolution Equations & Control Theory, 2014, 3 (1) : 191-206. doi: 10.3934/eect.2014.3.191

[19]

Daoyuan Fang, Bin Han, Matthias Hieber. Local and global existence results for the Navier-Stokes equations in the rotational framework. Communications on Pure & Applied Analysis, 2015, 14 (2) : 609-622. doi: 10.3934/cpaa.2015.14.609

[20]

Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085

2020 Impact Factor: 1.833

Article outline

[Back to Top]