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September  2020, 25(9): 3437-3460. doi: 10.3934/dcdsb.2020068

## Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization

 1 Department of Basic Science, Jilin Jianzhu University, Changchun 130118, China 2 School of Mathematics and Statistics Science, Ludong University, Yantai 264025, China

* Corresponding author

Received  June 2019 Revised  October 2019 Published  September 2020 Early access  April 2020

We consider an initial-boundary value problem for the incompressible four-component Keller-Segel-Navier-Stokes system with rotational flux
 \begin{align} \left\{ \begin{array}{l} n_t+u\cdot\nabla n = \Delta n-\nabla\cdot(nS(x,n,c)\nabla c)-nm,\quad x\in \Omega, t>0,\\ c_t+u\cdot\nabla c = \Delta c-c+m,\quad x\in \Omega, t>0,\\ m_t+u\cdot\nabla m = \Delta m-nm,\quad x\in \Omega, t>0, \qquad \qquad \qquad \qquad \qquad \qquad (*)\\ u_t+\kappa(u \cdot \nabla)u+\nabla P = \Delta u+(n+m)\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u = 0,\quad x\in \Omega, t>0\\ \end{array}\right. \end{align}
in a bounded domain
 $\Omega\subset \mathbb{R}^3$
with smooth boundary, where
 $\kappa\in \mathbb{R}$
is given constant,
 $S$
is a matrix-valued sensitivity satisfying
 $|S(x,n,c)|\leq C_S(1+n)^{-\alpha}$
with some
 $C_S> 0$
and
 $\alpha\geq 0$
. As the case
 $\kappa = 0$
(with
 $\alpha\geq\frac{1}{3}$
or the initial data satisfy a certain smallness condition) has been considered in [18], based on new gradient-like functional inequality, it is shown in the present paper that the corresponding initial-boundary problem with
 $\kappa \neq 0$
admits at least one global weak solution if
 $\alpha>0$
. To the best of our knowledge, this is the first analytical work for the full three-dimensional four-component chemotaxis-Navier-Stokes system.
Citation: Ling Liu, Jiashan Zheng, Gui Bao. Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3437-3460. doi: 10.3934/dcdsb.2020068
##### References:
 [1] N. Bellomo, A. Belloquid, 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. [2] X. Cao, Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with logistic source, J. Math. Anal. Appl., 412 (2014), 181-188.  doi: 10.1016/j.jmaa.2013.10.061. [3] M. Chae, K. 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. [4] R. Duan, A. Lorz and P. A. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673.  doi: 10.1080/03605302.2010.497199. [5] E. 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. [6] E. 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. [7] Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier–Stokes system, J. Differential Equations, 61 (1986), 186-212.  doi: 10.1016/0022-0396(86)90096-3. [8] J. Gu and F. Meng, Some new nonlinear Volterra-Fredholm type dynamic integral inequalities on time scales, Appl. Math. Comput., 245 (2014), 235-242.  doi: 10.1016/j.amc.2014.07.056. [9] P. He, Y. Wang and L. Zhao, A further study on a 3D chemotaxis-Stokes system with tensor-valued sensitivity, Appl. Math. Lett., 90 (2019), 23-29.  doi: 10.1016/j.aml.2018.09.019. [10] T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3. [11] D. Horstmann, From 1970 until present: the Keller–Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. [12] S. Ishida, Global existence and boundedness for chemotaxis–Navier–Stokes system with position-dependent sensitivity in 2D bounded domains, Discrete Contin. Dyn. Syst., 32 (2015), 3463-3482.  doi: 10.3934/dcds.2015.35.3463. [13] S. Ishida, K. Seki and T Yokota, Boundedness in quasilinear Keller–Segel systems of parabolic–parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028. [14] Y. Ke and J. Zheng, An optimal result for global existence in a three-dimensional Keller–Segel–Navier–Stokes system involving tensor-valued sensitivity with saturation, Calc. Var. Partial Differential Equations, 58 (2019), 27pp. doi: 10.1007/s00526-019-1568-2. [15] E. Keller and L. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6. [16] 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. [17] A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and efficiency of biological reactions: The critical reaction case, J. Math. Phys., 53 (2012), 9pp. doi: 10.1063/1.4742858. [18] J. Li, P. Pang and Y. Wang, Global boundedness and decay property of a three-dimensional Keller–Segel–Stokes system modeling coral fertilization, Nonlinearity, 32 (2019), 2815-2847.  doi: 10.1088/1361-6544/ab159b. [19] Y. Li and Y. Li, Finite-time blow-up in higher dimensional fully-parabolic chemotaxis system for two species, Nonlinear Anal., 109 (2014), 72-84.  doi: 10.1016/j.na.2014.05.021. [20] X. Li, Y. Wang and Z. Xiang, Global existence and boundedness in a 2D Keller-Segel-Stokes system with nonlinear diffusion and rotational flux, Commun. Math. Sci., 14 (2016), 1889-1910.  doi: 10.4310/CMS.2016.v14.n7.a5. [21] J.-G. Liu and A. Lorz, A coupled chemotaxis–fluid model: Global existence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 643-652.  doi: 10.1016/j.anihpc.2011.04.005. [22] J. Liu and Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations, 262 (2017), 5271-5305.  doi: 10.1016/j.jde.2017.01.024. [23] A. Lorz, Coupled chemotaxis fluid equations, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.  doi: 10.1142/S0218202510004507. [24] K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. [25] Y. Peng and Z. Xiang, Global existence and boundedness in a 3D Keller–Segel–Stokes system with nonlinear diffusion and rotational flux, Z. Angew. Math. Phys., 68 (2017), 26pp. doi: 10.1007/s00033-017-0816-6. [26] B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. doi: 10.1007/978-3-7643-7842-4. [27] J. Simon, Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl., 146 (1986), 65-96.  doi: 10.1007/BF01762360. [28] H. Sohr, The Navier–Stokes Equations. An Elementary Functional Analytic Approach, Birkhäuser Advanced Texts: Basel Textbooks, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8255-2. [29] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019. [30] Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis–Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178.  doi: 10.1016/j.anihpc.2012.07.002. [31] Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis–fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.  doi: 10.1007/s00033-015-0541-y. [32] J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003. [33] I. Tuval, L. Cisneros and C. Dombrowski, et al., Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277–2282. doi: 10.1073/pnas.0406724102. [34] Y. Wang and X. Cao, Global classical solutions of a 3D chemotaxis–Stokes system with rotation, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3235-3254.  doi: 10.3934/dcdsb.2015.20.3235. [35] Y. Wang, L. Liu, X. Zhang and Y. Wu, Positive solutions of an abstract fractional semipositone differential system model for bioprocesses of HIV infection, Applied Math. Comput., 258 (2015), 312-324.  doi: 10.1016/j.amc.2015.01.080. [36] Y. Wang and Z. Xiang, Global existence and boundedness in a Keller–Segel–Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations, 259 (2015), 7578-7609.  doi: 10.1016/j.jde.2015.08.027. [37] 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, J. Differential Equations, 261 (2016), 4944-4973.  doi: 10.1016/j.jde.2016.07.010. [38] Y. Wang, M. Winkler and Z. Xiang, Global classical solutions in a two-dimensional chemotaxis–Navier–Stokes system with subcritical sensitivity, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (2018), 421-466. [39] M. Winkler, Boundedness in the higher-dimensional parabolic–parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426. [40] 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. [41] M. Winkler, Stabilization in a two-dimensional chemotaxis–Navier–Stokes system, Arch. Ration. Mech. Anal., 211 (2014), 455-487.  doi: 10.1007/s00205-013-0678-9. [42] M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis–Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differential Equations, 54 (2015), 3789-3828.  doi: 10.1007/s00526-015-0922-2. [43] 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. [44] M. Winkler, Does fluid interaction affect regularity in the three-dimensional Keller-Segel System with saturated sensitivity?, J. Math. Fluid Mech., 20 (2018), 1889-1909.  doi: 10.1007/s00021-018-0395-0. [45] M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045. [46] F. Xu and L. Liu, On the well-posedness of the incompressible flow in porous media, J. Nonlinear Sci. Appl., 9 (2016), 6371-6381.  doi: 10.22436/jnsa.009.12.37. [47] C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial populations, SIAM J. Appl. Math., 70 (2009), 133-167.  doi: 10.1137/070711505. [48] Q. Zhang and X. Zheng, Global well-posedness for the two-dimensional incompressible chemotaxis–Navier–Stokes equations, SIAM J. Math. Anal., 46 (2014), 3078-3105.  doi: 10.1137/130936920. [49] J. Zheng, Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source, J. Differential Equations, 259 (2015), 120-140.  doi: 10.1016/j.jde.2015.02.003. [50] J. Zheng, Boundedness of solutions to a quasilinear parabolic–parabolic Keller–Segel system with a logistic source, J. Math. Anal. Appl., 431 (2015), 867-888.  doi: 10.1016/j.jmaa.2015.05.071. [51] J. Zheng, A note on boundedness of solutions to a higher-dimensional quasi-linear chemotaxis system with logistic source, ZAMM Z. Angew. Math. Mech., 97 (2017), 414-421.  doi: 10.1002/zamm.201600166. [52] J. Zheng, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 263 (2017), 2606-2629.  doi: 10.1016/j.jde.2017.04.005. [53] J. Zheng, Boundedness in a two-species quasi-linear chemotaxis system with two chemicals, Topol. Methods Nonlinear Anal., 49 (2017), 463-480.  doi: 10.12775/TMNA.2016.082. [54] J. Zheng, An optimal result for global existence and boundedness in a three-dimensional Keller-Segel-Stokes system with nonlinear diffusion, J. Differential Equations, 267 (2019), 2385-2415.  doi: 10.1016/j.jde.2019.03.013. [55] J. Zheng, A new result for the global boundedness and decay property of a three-dimensional Keller-Segel-Stokes system modeling coral fertilization, preprint. [56] J. Zheng, et. al., A new result for global existence and boundedness of solutions to a parabolic–parabolic Keller–Segel system with logistic source, J. Math. Anal. Appl., 462 (2018), 1–25. doi: 10.1016/j.jmaa.2018.01.064. [57] J. Zheng and Y. Wang, A note on global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 669-686.  doi: 10.3934/dcdsb.2017032.

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##### References:
 [1] N. Bellomo, A. Belloquid, 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. [2] X. Cao, Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with logistic source, J. Math. Anal. Appl., 412 (2014), 181-188.  doi: 10.1016/j.jmaa.2013.10.061. [3] M. Chae, K. 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. [4] R. Duan, A. Lorz and P. A. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673.  doi: 10.1080/03605302.2010.497199. [5] E. 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. [6] E. 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. [7] Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier–Stokes system, J. Differential Equations, 61 (1986), 186-212.  doi: 10.1016/0022-0396(86)90096-3. [8] J. Gu and F. Meng, Some new nonlinear Volterra-Fredholm type dynamic integral inequalities on time scales, Appl. Math. Comput., 245 (2014), 235-242.  doi: 10.1016/j.amc.2014.07.056. [9] P. He, Y. Wang and L. Zhao, A further study on a 3D chemotaxis-Stokes system with tensor-valued sensitivity, Appl. Math. Lett., 90 (2019), 23-29.  doi: 10.1016/j.aml.2018.09.019. [10] T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3. [11] D. Horstmann, From 1970 until present: the Keller–Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. [12] S. Ishida, Global existence and boundedness for chemotaxis–Navier–Stokes system with position-dependent sensitivity in 2D bounded domains, Discrete Contin. Dyn. Syst., 32 (2015), 3463-3482.  doi: 10.3934/dcds.2015.35.3463. [13] S. Ishida, K. Seki and T Yokota, Boundedness in quasilinear Keller–Segel systems of parabolic–parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028. [14] Y. Ke and J. Zheng, An optimal result for global existence in a three-dimensional Keller–Segel–Navier–Stokes system involving tensor-valued sensitivity with saturation, Calc. Var. Partial Differential Equations, 58 (2019), 27pp. doi: 10.1007/s00526-019-1568-2. [15] E. Keller and L. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6. [16] 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. [17] A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and efficiency of biological reactions: The critical reaction case, J. Math. Phys., 53 (2012), 9pp. doi: 10.1063/1.4742858. [18] J. Li, P. Pang and Y. Wang, Global boundedness and decay property of a three-dimensional Keller–Segel–Stokes system modeling coral fertilization, Nonlinearity, 32 (2019), 2815-2847.  doi: 10.1088/1361-6544/ab159b. [19] Y. Li and Y. Li, Finite-time blow-up in higher dimensional fully-parabolic chemotaxis system for two species, Nonlinear Anal., 109 (2014), 72-84.  doi: 10.1016/j.na.2014.05.021. [20] X. Li, Y. Wang and Z. Xiang, Global existence and boundedness in a 2D Keller-Segel-Stokes system with nonlinear diffusion and rotational flux, Commun. Math. Sci., 14 (2016), 1889-1910.  doi: 10.4310/CMS.2016.v14.n7.a5. [21] J.-G. Liu and A. Lorz, A coupled chemotaxis–fluid model: Global existence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 643-652.  doi: 10.1016/j.anihpc.2011.04.005. [22] J. Liu and Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations, 262 (2017), 5271-5305.  doi: 10.1016/j.jde.2017.01.024. [23] A. Lorz, Coupled chemotaxis fluid equations, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.  doi: 10.1142/S0218202510004507. [24] K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. [25] Y. Peng and Z. Xiang, Global existence and boundedness in a 3D Keller–Segel–Stokes system with nonlinear diffusion and rotational flux, Z. Angew. Math. Phys., 68 (2017), 26pp. doi: 10.1007/s00033-017-0816-6. [26] B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. doi: 10.1007/978-3-7643-7842-4. [27] J. Simon, Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl., 146 (1986), 65-96.  doi: 10.1007/BF01762360. [28] H. Sohr, The Navier–Stokes Equations. An Elementary Functional Analytic Approach, Birkhäuser Advanced Texts: Basel Textbooks, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8255-2. [29] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019. [30] Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis–Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178.  doi: 10.1016/j.anihpc.2012.07.002. [31] Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis–fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.  doi: 10.1007/s00033-015-0541-y. [32] J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003. [33] I. Tuval, L. Cisneros and C. Dombrowski, et al., Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277–2282. doi: 10.1073/pnas.0406724102. [34] Y. Wang and X. Cao, Global classical solutions of a 3D chemotaxis–Stokes system with rotation, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3235-3254.  doi: 10.3934/dcdsb.2015.20.3235. [35] Y. Wang, L. Liu, X. Zhang and Y. Wu, Positive solutions of an abstract fractional semipositone differential system model for bioprocesses of HIV infection, Applied Math. Comput., 258 (2015), 312-324.  doi: 10.1016/j.amc.2015.01.080. [36] Y. Wang and Z. Xiang, Global existence and boundedness in a Keller–Segel–Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations, 259 (2015), 7578-7609.  doi: 10.1016/j.jde.2015.08.027. [37] 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, J. Differential Equations, 261 (2016), 4944-4973.  doi: 10.1016/j.jde.2016.07.010. [38] Y. Wang, M. Winkler and Z. Xiang, Global classical solutions in a two-dimensional chemotaxis–Navier–Stokes system with subcritical sensitivity, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (2018), 421-466. [39] M. Winkler, Boundedness in the higher-dimensional parabolic–parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426. [40] M. 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