# American Institute of Mathematical Sciences

August  2022, 42(8): 4095-4125. doi: 10.3934/dcds.2022047

## Boundedness and stabilization of a three-dimensional parabolic-elliptic Keller-Segel-Stokes system

 School of Mathematics and Information Sciences, Yantai University, Yantai 264005, China

*Corresponding author: Jiashan Zheng

Received  October 2021 Revised  March 2022 Published  August 2022 Early access  April 2022

Fund Project: The work is supported by NSF grant 11601215

This paper is concerned with the volume-filling effect on global solvability and stabilization in a parabolic-elliptic Keller-Segel-Stokes systems
 \begin{align} \left\{ \begin{array}{l} n_t+u\cdot\nabla n = \Delta n-\nabla\cdot(nS(n)\nabla c),\quad x\in \Omega, t>0,\\ u\cdot\nabla c = \Delta c-c+n,\quad x\in \Omega, t>0,\\ u_t+\nabla P = \Delta u+n\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u = 0,\quad x\in \Omega, t>0\\ \end{array}\right. \end{align} \;\;\;\;\;\;\;\;\;\;\;\;(KSF)
with no-flux boundary conditions for
 $n$
and
 $c$
as well as no-slip boundary condition for
 $u$
in a bounded domain
 $\Omega \subseteq \mathbb{R}^3$
with smooth boundary. Here the nonnegative function
 $S\in C^2(\bar{\Omega})$
denotes the chemotactic sensitivity which fulfills
 $|S(n)|\leq C_S(1 + n)^{-\alpha} \; \; \; \; \text{for all}\; \; n\geq0$
with some
 $C_S > 0$
and
 $\alpha> 0$
. Imposing no restriction on the size of the initial data, by seeking some new functionals and using the bootstrap arguments on the system, we establish the existence and boundedness of global classical solutions to parabolic-elliptic Keller-Segel-Stokes system under the assumption
 $\alpha> \frac{1}{2}$
. On the basis of this, we further prove that if the chemotactic coefficient
 $C_S$
is appropriately small, the obtained solutions are shown to approach the spatially homogeneous steady state
 $(\bar{n}_0, \bar{n}_0, 0)$
in the large time limit, where
 $\bar{n}_0 = \frac{1}{|\Omega|}\int_{\Omega}n_0$
, provided that merely
 $n_0\not \equiv0$
on
 $\Omega$
.
Citation: Pengmei Zhang, Jiashan Zheng. Boundedness and stabilization of a three-dimensional parabolic-elliptic Keller-Segel-Stokes system. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 4095-4125. doi: 10.3934/dcds.2022047
##### 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. [2] R. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673.  doi: 10.1080/03605302.2010.497199. [3] 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. [4] K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. [5] 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. [6] H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8255-2. [7] 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. [8] 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. (5), 18 (2018), 421-466.  doi: 10.2422/2036-2145.201603_004. [9] 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. [10] 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. [11] 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. [12] 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. [13] 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. [14] M. Winkler, Global existence and stabilization in a degenerate chemotaxis–Stokes system with mildly strong diffusion enhancement, J. Differential Equations, 264 (2018), 6109-6151.  doi: 10.1016/j.jde.2018.01.027. [15] 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. [16] 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. [17] 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. [18] M. Winkler, A three-dimensional Keller–Segel–Navier–Stokes system with logistic source: Global weak solutions and asymptotic stabilization, J. Funct. Anal., 276 (2019), 1339-1401.  doi: 10.1016/j.jfa.2018.12.009. [19] 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. [20] J. Zheng, Eventual smoothness and stabilization in a three-dimensional Keller-Segel-Navier-Stokes system with rotational flux, Calc. Var. Partial Differential Equations, 61 (2022), 34pp. doi: 10.1007/s00526-021-02164-6. [21] J. Zheng, Global classical solutions and stabilization in a two-dimensional parabolic–elliptic Keller–Segel–Stokes system, J. Math. Fluid Mech., 23 (2021), 25pp. doi: 10.1007/s00021-021-00600-3. [22] J. Zheng, Global existence and boundedness in a three-dimensional chemotaxis–Stokes system with nonlinear diffusion and general sensitivity, Ann. Mat. Pura Appl. (4), 201 (2022), 243-288.  doi: 10.1007/s10231-021-01115-4. [23] J. Zheng, A new result for the global existence (and boundedness) and regularity of a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization, J. Differential Equations, 272 (2021), 164-202.  doi: 10.1016/j.jde.2020.09.029. [24] 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. [25] J. Zheng and Y. Ke, Blow-up prevention by nonlinear diffusion in a 2D Keller–Segel–Navier–Stokes system with rotational flux, J. Differential Equations, 268 (2020), 7092-7120.  doi: 10.1016/j.jde.2019.11.071. [26] J. Zheng and Y. Ke, Global bounded weak solutions for a chemotaxis–Stokes system with nonlinear diffusion and rotation, J. Differential Equations, 289 (2021), 182-235.  doi: 10.1016/j.jde.2021.04.020.

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##### 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. [2] R. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673.  doi: 10.1080/03605302.2010.497199. [3] 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. [4] K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. [5] 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. [6] H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8255-2. [7] 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. [8] 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. (5), 18 (2018), 421-466.  doi: 10.2422/2036-2145.201603_004. [9] 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. [10] 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. [11] 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. [12] 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. [13] 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. [14] M. Winkler, Global existence and stabilization in a degenerate chemotaxis–Stokes system with mildly strong diffusion enhancement, J. Differential Equations, 264 (2018), 6109-6151.  doi: 10.1016/j.jde.2018.01.027. [15] 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. [16] 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. [17] 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. [18] M. Winkler, A three-dimensional Keller–Segel–Navier–Stokes system with logistic source: Global weak solutions and asymptotic stabilization, J. Funct. Anal., 276 (2019), 1339-1401.  doi: 10.1016/j.jfa.2018.12.009. [19] 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. [20] J. Zheng, Eventual smoothness and stabilization in a three-dimensional Keller-Segel-Navier-Stokes system with rotational flux, Calc. Var. Partial Differential Equations, 61 (2022), 34pp. doi: 10.1007/s00526-021-02164-6. [21] J. Zheng, Global classical solutions and stabilization in a two-dimensional parabolic–elliptic Keller–Segel–Stokes system, J. Math. Fluid Mech., 23 (2021), 25pp. doi: 10.1007/s00021-021-00600-3. [22] J. Zheng, Global existence and boundedness in a three-dimensional chemotaxis–Stokes system with nonlinear diffusion and general sensitivity, Ann. Mat. Pura Appl. (4), 201 (2022), 243-288.  doi: 10.1007/s10231-021-01115-4. [23] J. Zheng, A new result for the global existence (and boundedness) and regularity of a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization, J. Differential Equations, 272 (2021), 164-202.  doi: 10.1016/j.jde.2020.09.029. [24] 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. [25] J. Zheng and Y. Ke, Blow-up prevention by nonlinear diffusion in a 2D Keller–Segel–Navier–Stokes system with rotational flux, J. Differential Equations, 268 (2020), 7092-7120.  doi: 10.1016/j.jde.2019.11.071. [26] J. Zheng and Y. Ke, Global bounded weak solutions for a chemotaxis–Stokes system with nonlinear diffusion and rotation, J. Differential Equations, 289 (2021), 182-235.  doi: 10.1016/j.jde.2021.04.020.
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