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doi: 10.3934/era.2021050
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## 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. Bellomo, A. Bellouquid, Y. 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. Cao, S. 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 Passo, H. 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. Fujie, A. 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. Hirata, S. Kurima, M. 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. Stinner, C. 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. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. 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. Bellomo, A. Bellouquid, Y. 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. Cao, S. 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 Passo, H. 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. Fujie, A. 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. Hirata, S. Kurima, M. 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. Stinner, C. 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. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. 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
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