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February 2019, 24(2): 423-447. doi: 10.3934/dcdsb.2018180

Global existence for an attraction-repulsion chemotaxis fluid model with logistic source

 1 Universidad Industrial de Santander, Escuela de Matemáticas, Bucaramanga, A.A. 678, Colombia 2 Universidade Estadual de Campinas, Departamento de Matemática-IMECC, CEP 13083-859, Campinas-SP, Brazil

* Corresponding author: Élder J. Villamizar-Roa

Received  November 2017 Revised  January 2018 Published  June 2018

Fund Project: The second author has been partially supported by CNPq and FAPESP, Brazil. The third author has been supported by Vicerrectoría de Investigación y Extensión of Universidad Industrial de Santander, and Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco José de Caldas, contrato Colciencias FP 44842-157-2016. The authors would like to thank an anonymous referee for useful remarks and suggestions

We consider an attraction-repulsion chemotaxis model coupled with the Navier-Stokes system. This model describes the interaction between a type of cells (e.g., bacteria), which proliferate following a logistic law, and two chemical signals produced by the cells themselves that degraded at a constant rate. Also, it is considered that the chemoattractant is consumed with a rate proportional to the amount of organisms. The cells and chemical substances are transported by a viscous incompressible fluid under the influence of a force due to the aggregation of cells. We prove the existence of global mild solutions in bounded domains of $\mathbb{R}^N,$ $N = 2, 3,$ for small initial data in $L^p$-spaces.

Citation: Abelardo Duarte-Rodríguez, Lucas C. F. Ferreira, Élder J. Villamizar-Roa. Global existence for an attraction-repulsion chemotaxis fluid model with logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 423-447. doi: 10.3934/dcdsb.2018180
References:
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Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities, Calc. Var. Partial Differential Equations, 55 (2016), Art. 107, 39 pp. doi: 10.1007/s00526-016-1027-2. [6] T. Cazenave and F. B. Weissler, Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations, Mathematische Zeitschrift, 228 (1998), 83-120. doi: 10.1007/PL00004606. [7] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, The International Series of Monographs on Physics Clarendon Press, Oxford, 1961. [8] M.A. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Mathematical Models and Methods in Applied Sciences, 15 (2005), 1685-1734. doi: 10.1142/S0218202505000947. [9] M. A. Chaplain and A. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor, Mathematical Medicine and Biology, 10 (1993), 149-168. doi: 10.1093/imammb/10.3.149. [10] A. Chertock, K. Fellner, A. Kurganov, A. Lorz and P. Markowich, Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach, Journal of Fluid Mechanics, 694 (2012), 155-190. doi: 10.1017/jfm.2011.534. [11] H. J. Choe and B. Lkhagvasuren, Global existence result for chemotaxis Navier-Stokes equations in the critical Besov spaces, Journal of Mathematical Analysis and Applications, 446 (2017), 1415-1426. doi: 10.1016/j.jmaa.2016.09.050. [12] C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein and J. O. Kessler, Self-concentration and large-scale coherence in bacterial dynamics, Physical Review Letters, 93 (2004), 98-103. [13] R. Duan and Z. Xiang, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, International Mathematics Research Notices, 2014 (2012), 1833-1852. doi: 10.1093/imrn/rns270. [14] E. Espejo and T. Suzuki, Reaction terms avoiding aggregation in slow fluids, Nonlinear Analysis: Real World Applications, 21 (2015), 110-126. doi: 10.1016/j.nonrwa.2014.07.001. [15] L. C. F. Ferreira and E. J. Villamizar-Roa, Self-similar solutions, uniqueness and long-time asymptotic behavior for semilinear heat equations, Differential and Integral Equations, 19 (2006), 1349-1370. [16] L. C. F. Ferreira and E. J. Villamizar-Roa, Well-posedness and asymptotic behaviour for the convection problem, Nonlinearity, 19 (2006), 2169-2191. doi: 10.1088/0951-7715/19/9/011. [17] D. Fujiwara and H. Morimoto, An Lr-theorem of the Helmholtz decomposition of vector fields, IA Math, 24 (1977), 685-700. [18] Y. Giga, Analyticity of the semigroup generated by the Stokes operator in Lr spaces, Mathematische Zeitschrift, 178 (1981), 297-329. doi: 10.1007/BF01214869. [19] N. A. Hill and T. J. Pedley, Bioconvection, Fluid Dynamics Research, 37 (2005), 1-20. doi: 10.1016/j.fluiddyn.2005.03.002. [20] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. [21] T. Hillen, K. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Mathematical Models and Methods in Applied Sciences, 23 (2013), 165-198. doi: 10.1142/S0218202512500480. [22] D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, Journal of Nonlinear Science, 21 (2011), 231-270. doi: 10.1007/s00332-010-9082-x. [23] S. Ishida, Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains, Discrete & Continuous Dynamical Systems-A, 35 (2015), 3463-3482. doi: 10.3934/dcds.2015.35.3463. [24] J. Jiang, H. Wu and S. Zheng, Global existence and asymptotic behavior of solutions to a chemotaxis-fluid system on general bounded domains, Asymptotic Analysis, 92 (2015), 249-258. [25] T. Kato, Strong Lp-solutions of the Navier-Stokes equation in Rm with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182. [26] T. Kato, Strong solutions of the Navier-Stokes equation in Morrey spaces, Bol. Soc. Brasil. Mat., 22 (1992), 127-155. doi: 10.1007/BF01232939. [27] A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and enhancement of biological reactions, Communications in Partial Differential Equations, 37 (2012), 298-318. doi: 10.1080/03605302.2011.589879. [28] H. Kozono, M. Miura and Y. Sugiyama, Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid, Journal of Functional Analysis, 270 (2016), 1663-1683. doi: 10.1016/j.jfa.2015.10.016. [29] J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, Journal of Differential Equations, 258 (2015), 1158-1191. doi: 10.1016/j.jde.2014.10.016. [30] J. Lankeit, Long-term behaviour in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci., 26 (2016), 2071-2109. doi: 10.1142/S021820251640008X. [31] D. Li, C. Mu, K. Lin and L. Wang, Large time behavior of solution to an attraction-repulsion chemotaxis system with logistic source in three dimensions, Journal of Mathematical Analysis and Applications, 448 (2016), 914-936. doi: 10.1016/j.jmaa.2016.11.036. [32] X. Li, Boundedness in a two-dimensional attraction-repulsion system with nonlinear diffusion, Mathematical Methods in the Applied Sciences, 39 (2016), 289-301. doi: 10.1002/mma.3477. [33] X. Li and Z. Xiang, On an attraction-repulsion chemotaxis system with a logistic source, IMA Journal of Applied Mathematics, 81 (2016), 165-198. doi: 10.1093/imamat/hxv033. [34] J. Liu and Y. Wang, Global existence and boundedness in a Keller-Segel-(Navier-) Stokes system with signal-dependent sensitivity, Journal of Mathematical Analysis and Applications, 447 (2017), 499-528. doi: 10.1016/j.jmaa.2016.10.028. [35] J. Liu and Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system involving a tensor-valued sensitivity with saturation, Journal of Differential Equations, 262 (2017), 5271-5305. doi: 10.1016/j.jde.2017.01.024. [36] P. Liu, J. Shi and Z.-A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625. doi: 10.3934/dcdsb.2013.18.2597. [37] M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: Is there a connection?, Bulletin of Mathematical Biology, 65 (2003), 693-730. doi: 10.1016/S0092-8240(03)00030-2. [38] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser/Springer Basel AG, Basel, 1995. [39] N. V. Mantzaris, S. Webb and H. G. Othmer, Mathematical modeling of tumor-induced angiogenesis, Journal of Mathematical Biology, 49 (2004), 111-187. doi: 10.1007/s00285-003-0262-2. [40] X. Mora, Semilinear parabolic problems define semiflows on Ck spaces, Transactions of the American Mathematical Society, 278 (1983), 21-55. doi: 10.2307/1999300. [41] A. Quinlan and B. Straughan, Decay bounds in a model for aggregation of microglia: Application to Alzheimer's disease senile plaques, Proceedings of the Royal Society A, 461 (2005), 2887-2897. doi: 10.1098/rspa.2005.1483. [42] Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 30 (2013), 157-178. doi: 10.1016/j.anihpc.2012.07.002. [43] Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Zeitschrift für angewandte Mathematik und Physik, 66 (2015), 2555-2573. doi: 10.1007/s00033-015-0541-y. [44] Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system Z. Angew. Math. Phys., 67 (2016), Art. 138, 23 pp. doi: 10.1007/s00033-016-0732-1. [45] J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Communications in Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003. [46] I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proceedings of the National Academy of Sciences of the United States of America, 102 (2005), 2277-2282. doi: 10.1073/pnas.0406724102. [47] R. Tyson, S. R. Lubkin and J. D. Murray, Model and analysis of chemotactic bacterial patterns in a liquid medium, Journal of Mathematical Biology, 38 (1999), 359-375. doi: 10.1007/s002850050153. [48] Y. Wang, Boundedness in a three-dimensional attraction-repulsion chemotaxis system with nonlinear diffusion and logistic source, Electronic Journal of Differential Equations, 176 (2016), 1-21. [49] Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation: The 3D case, Journal of Differential Equations, 261 (2016), 4944-4973. doi: 10.1016/j.jde.2016.07.010. [50] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008. [51] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Communications in Partial Differential Equations, 35 (2010), 1516-1537. doi: 10.1080/03605300903473426. [52] M. Winkler, Global large-data solutions in a chemotaxis-(Navier-) Stokes system modeling cellular swimming in fluid drops, Communications in Partial Differential Equations, 37 (2012), 319-351. doi: 10.1080/03605302.2011.591865. [53] M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, Journal of Differential Equations, 257 (2014), 1056-1077. doi: 10.1016/j.jde.2014.04.023. [54] M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calculus of Variations and Partial Differential Equations, 54 (2015), 3789-3828. doi: 10.1007/s00526-015-0922-2. [55] M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 33 (2016), 1329-1352. doi: 10.1016/j.anihpc.2015.05.002. [56] D. Woodward, R. Tyson, M. Myerscough, J. D. Murray, E. Budrene, and H. Berg, Spatio-temporal patterns generated by Salmonella typhimurium Biophysical Journal, 68 (1995), no. 5, 2181. doi: 10.1016/S0006-3495(95)80400-5. [57] Q. Zhang and Y. 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References:
 [1] M. Aida, K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Chemotaxis and growth system with singular sensitivity function, Nonlinear Analysis: Real World Applications, 6 (2005), 323-336. doi: 10.1016/j.nonrwa.2004.08.011. [2] L. Angiuli, D. Pallara and F. Y. Paronetto, Analytic semigroups generated in L1 by second order elliptic operators via duality methods, Semigroup Forum, Springer, 80 (2010), 255-271. doi: 10.1007/s00233-009-9200-y. [3] N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Mathematical Models and Methods in Applied Sciences, 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X. [4] M. Braukhoff, Global (weak) solution of the chemotaxis-Navier-Stokes equations with non-homogeneous boundary conditions and logistic growth, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 34 (2017), 1013-1039. doi: 10.1016/j.anihpc.2016.08.003. [5] X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities, Calc. Var. Partial Differential Equations, 55 (2016), Art. 107, 39 pp. doi: 10.1007/s00526-016-1027-2. [6] T. Cazenave and F. B. Weissler, Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations, Mathematische Zeitschrift, 228 (1998), 83-120. doi: 10.1007/PL00004606. [7] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, The International Series of Monographs on Physics Clarendon Press, Oxford, 1961. [8] M.A. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Mathematical Models and Methods in Applied Sciences, 15 (2005), 1685-1734. doi: 10.1142/S0218202505000947. [9] M. A. Chaplain and A. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor, Mathematical Medicine and Biology, 10 (1993), 149-168. doi: 10.1093/imammb/10.3.149. [10] A. Chertock, K. Fellner, A. Kurganov, A. Lorz and P. Markowich, Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach, Journal of Fluid Mechanics, 694 (2012), 155-190. doi: 10.1017/jfm.2011.534. [11] H. J. Choe and B. Lkhagvasuren, Global existence result for chemotaxis Navier-Stokes equations in the critical Besov spaces, Journal of Mathematical Analysis and Applications, 446 (2017), 1415-1426. doi: 10.1016/j.jmaa.2016.09.050. [12] C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein and J. O. Kessler, Self-concentration and large-scale coherence in bacterial dynamics, Physical Review Letters, 93 (2004), 98-103. [13] R. Duan and Z. Xiang, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, International Mathematics Research Notices, 2014 (2012), 1833-1852. doi: 10.1093/imrn/rns270. [14] E. Espejo and T. Suzuki, Reaction terms avoiding aggregation in slow fluids, Nonlinear Analysis: Real World Applications, 21 (2015), 110-126. doi: 10.1016/j.nonrwa.2014.07.001. [15] L. C. F. Ferreira and E. J. Villamizar-Roa, Self-similar solutions, uniqueness and long-time asymptotic behavior for semilinear heat equations, Differential and Integral Equations, 19 (2006), 1349-1370. [16] L. C. F. Ferreira and E. J. Villamizar-Roa, Well-posedness and asymptotic behaviour for the convection problem, Nonlinearity, 19 (2006), 2169-2191. doi: 10.1088/0951-7715/19/9/011. [17] D. Fujiwara and H. Morimoto, An Lr-theorem of the Helmholtz decomposition of vector fields, IA Math, 24 (1977), 685-700. [18] Y. Giga, Analyticity of the semigroup generated by the Stokes operator in Lr spaces, Mathematische Zeitschrift, 178 (1981), 297-329. doi: 10.1007/BF01214869. [19] N. A. Hill and T. J. Pedley, Bioconvection, Fluid Dynamics Research, 37 (2005), 1-20. doi: 10.1016/j.fluiddyn.2005.03.002. [20] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. [21] T. Hillen, K. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Mathematical Models and Methods in Applied Sciences, 23 (2013), 165-198. doi: 10.1142/S0218202512500480. [22] D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, Journal of Nonlinear Science, 21 (2011), 231-270. doi: 10.1007/s00332-010-9082-x. [23] S. Ishida, Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains, Discrete & Continuous Dynamical Systems-A, 35 (2015), 3463-3482. doi: 10.3934/dcds.2015.35.3463. [24] J. Jiang, H. Wu and S. Zheng, Global existence and asymptotic behavior of solutions to a chemotaxis-fluid system on general bounded domains, Asymptotic Analysis, 92 (2015), 249-258. [25] T. Kato, Strong Lp-solutions of the Navier-Stokes equation in Rm with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182. [26] T. Kato, Strong solutions of the Navier-Stokes equation in Morrey spaces, Bol. Soc. Brasil. Mat., 22 (1992), 127-155. doi: 10.1007/BF01232939. [27] A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and enhancement of biological reactions, Communications in Partial Differential Equations, 37 (2012), 298-318. doi: 10.1080/03605302.2011.589879. [28] H. Kozono, M. Miura and Y. Sugiyama, Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid, Journal of Functional Analysis, 270 (2016), 1663-1683. doi: 10.1016/j.jfa.2015.10.016. [29] J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, Journal of Differential Equations, 258 (2015), 1158-1191. doi: 10.1016/j.jde.2014.10.016. [30] J. Lankeit, Long-term behaviour in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci., 26 (2016), 2071-2109. doi: 10.1142/S021820251640008X. [31] D. Li, C. Mu, K. Lin and L. Wang, Large time behavior of solution to an attraction-repulsion chemotaxis system with logistic source in three dimensions, Journal of Mathematical Analysis and Applications, 448 (2016), 914-936. doi: 10.1016/j.jmaa.2016.11.036. [32] X. Li, Boundedness in a two-dimensional attraction-repulsion system with nonlinear diffusion, Mathematical Methods in the Applied Sciences, 39 (2016), 289-301. doi: 10.1002/mma.3477. [33] X. Li and Z. Xiang, On an attraction-repulsion chemotaxis system with a logistic source, IMA Journal of Applied Mathematics, 81 (2016), 165-198. doi: 10.1093/imamat/hxv033. [34] J. Liu and Y. Wang, Global existence and boundedness in a Keller-Segel-(Navier-) Stokes system with signal-dependent sensitivity, Journal of Mathematical Analysis and Applications, 447 (2017), 499-528. doi: 10.1016/j.jmaa.2016.10.028. [35] J. Liu and Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system involving a tensor-valued sensitivity with saturation, Journal of Differential Equations, 262 (2017), 5271-5305. doi: 10.1016/j.jde.2017.01.024. [36] P. Liu, J. Shi and Z.-A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625. doi: 10.3934/dcdsb.2013.18.2597. [37] M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: Is there a connection?, Bulletin of Mathematical Biology, 65 (2003), 693-730. doi: 10.1016/S0092-8240(03)00030-2. [38] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser/Springer Basel AG, Basel, 1995. [39] N. V. Mantzaris, S. Webb and H. G. Othmer, Mathematical modeling of tumor-induced angiogenesis, Journal of Mathematical Biology, 49 (2004), 111-187. doi: 10.1007/s00285-003-0262-2. [40] X. Mora, Semilinear parabolic problems define semiflows on Ck spaces, Transactions of the American Mathematical Society, 278 (1983), 21-55. doi: 10.2307/1999300. [41] A. Quinlan and B. Straughan, Decay bounds in a model for aggregation of microglia: Application to Alzheimer's disease senile plaques, Proceedings of the Royal Society A, 461 (2005), 2887-2897. doi: 10.1098/rspa.2005.1483. [42] Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 30 (2013), 157-178. doi: 10.1016/j.anihpc.2012.07.002. [43] Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Zeitschrift für angewandte Mathematik und Physik, 66 (2015), 2555-2573. doi: 10.1007/s00033-015-0541-y. [44] Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system Z. Angew. Math. Phys., 67 (2016), Art. 138, 23 pp. doi: 10.1007/s00033-016-0732-1. [45] J. I. Tello and M. 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Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation: The 3D case, Journal of Differential Equations, 261 (2016), 4944-4973. doi: 10.1016/j.jde.2016.07.010. [50] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008. [51] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Communications in Partial Differential Equations, 35 (2010), 1516-1537. doi: 10.1080/03605300903473426. [52] M. Winkler, Global large-data solutions in a chemotaxis-(Navier-) Stokes system modeling cellular swimming in fluid drops, Communications in Partial Differential Equations, 37 (2012), 319-351. doi: 10.1080/03605302.2011.591865. [53] M. 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