Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity

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July 2018, 38(7): 3595-3616. doi: 10.3934/dcds.2018155

Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity

Hai-Yang Jin ^,

Department of Mathematics, South China University of Technology, Guangzhou 510640, China

* Corresponding author: Hai-Yang Jin

Received October 2017 Published April 2018

Fund Project: The research of H.Y. Jin was supported by the NSF of China No. 11501218, and the Fundamental Research Funds for the Central Universities (No. 2017MS107)

This paper is concerned with the following Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity

$\begin{cases}\tag{*}n_t+u·\nabla n = \nabla ·(d(c)\nabla n)-\nabla ·(χ (c) n\nabla c)+a n-bn^2, &x∈ Ω, ~~t>0, \\ c_t+u·\nabla c = Δ c+ n-c,&x∈ Ω, ~~t>0, \\ u_t+ u·\nabla u = Δ u-\nabla P+n\nabla φ,&x∈ Ω, ~~t>0, \\\nabla · u = 0& x∈ Ω, \ t>0, \end{cases}$

in a bounded smooth domain

$Ω\subset \mathbb{R}^2$

with homogeneous Neumann boundary conditions, where

$a≥0$

and

$b>0$

are constants, and the functions

$d(c)$

and

$χ(c)$

satisfy the following assumptions:

●

$(d(c), χ (c))∈ [C^2([0, ∞))]^2$

with

$d(c), χ(c)>0$

for all

$c≥0$

$d'(c)<0$

and

$\lim\limits_{c\to∞}d(c) = 0$

●

$\lim\limits_{c\to∞} \frac{χ (c)}{d(c)}$

and

$\lim\limits_{c\to∞}\frac{d'(c)}{d(c)}$

exist.

The difficulty in analysis of system (*) is the possible degeneracy of diffusion due to the condition

$\lim\limits_{c\to∞}d(c) = 0$

. In this paper, we will use function

$d(c)$

as weight function and employ the method of energy estimate to establish the global existence of classical solutions of (*) with uniform-in-time bound. Furthermore, by constructing a Lyapunov functional, we show that the global classical solution

$(n, c, u)$

will converge to the constant state

$(\frac{a}{b}, \frac{a}{b}, 0)$

$b>\frac{K_0}{16}$

with

$K_0 = \max\limits_{0≤c ≤∞}\frac{|χ(c)|^2}{d(c)}$

Keywords: Chemotaxis, boundedness, large time behavior, signal-dependent diffusion.

Mathematics Subject Classification: Primary: 35A01, 35B40, 35K55, 35Q92, 92C17.

Citation: Hai-Yang Jin. Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3595-3616. doi: 10.3934/dcds.2018155

References:

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[2]	J. P. Bourguignon and H. Brezis, Remarks on Euler Equation, J. Functional Analysis, 15 (1974), 341-363. doi: 10.1016/0022-1236(74)90027-5. Google Scholar
[3]	M. Chae, K. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297. doi: 10.3934/dcds.2013.33.2271. Google Scholar
[4]	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. Google Scholar
[5]	A. Chertock, K. Fellner, A. Kurganov, A. Lorz and P. A. Markowich, Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: A high-resolution numerical approach, J. Fluid Mech., 694 (2012), 155-190. doi: 10.1017/jfm.2011.534. Google Scholar
[6]	S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237. doi: 10.1016/0025-5564(81)90055-9. Google Scholar
[7]	T. Ciéslak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851. doi: 10.1016/j.jde.2012.01.045. Google Scholar
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[12]	M. Eisenbach, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion. Chemotaxis, Imperial College Press, London, 2004.Google Scholar
[13]	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. Google Scholar
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[19]	T. Hillen and A. Potapov, The one-dimensional chemotaxis model: Global existence and asymptotic profile, Math. Methods Appl. Sci., 27 (2004), 1783-1801. doi: 10.1002/mma.569. Google Scholar
[20]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅰ, Jahresber. Deutsch. Math.-Verien., 105 (2003), 103-165. Google Scholar
[21]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅱ, Jahresber. Deutsch. Math.-Verien., 106 (2004), 51-69. Google Scholar
[22]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022. Google Scholar
[23]	W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6. Google Scholar
[24]	H. Y. Jin, Y. J. Kim and Z. A. Wang, Boundedness, stabilization and pattern formation driven by density-suppressed motility, SIAM J. Appl. Math., 2018, to appear.Google Scholar
[25]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. Google Scholar
[26]	R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398. doi: 10.1016/j.jmaa.2008.01.005. Google Scholar
[27]	O. Ladyzhenskaya, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, RI, 1968. Google Scholar
[28]	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. Google Scholar
[29]	A. Lorz, Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004. doi: 10.1142/S0218202510004507. Google Scholar
[30]	A. Lorz, A coupled Keller-Segel-Stokes model: Global existence for small initial data and blow-up delay, Commun. Math. Sci., 10 (2012), 555-574. doi: 10.4310/CMS.2012.v10.n2.a7. Google Scholar
[31]	T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. Google Scholar
[32]	T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. Google Scholar
[33]	K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. Google Scholar
[34]	K. Painter and J. A. Sherratt, Modeling the movement of interacting cell populations, J. Theor. Biol., 225 (2003), 327-339. doi: 10.1016/S0022-5193(03)00258-3. Google Scholar
[35]	M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178. doi: 10.1006/jdeq.1993.1045. Google Scholar
[36]	T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367. doi: 10.4310/MAA.2001.v8.n2.a9. Google Scholar
[37]	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. Google Scholar
[38]	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. Google Scholar
[39]	Y. S. Tao and M. Winkler, Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250. doi: 10.1137/15M1014115. Google Scholar
[40]	Y. S. 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. Google Scholar
[41]	Y. S. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2016), 2555-2573. doi: 10.1007/s00033-015-0541-y. Google Scholar
[42]	Y. Tao and M. Winkler, Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system, Math. Models Meth. Appl. Sci., 27 (2017), 1645-1683. doi: 10.1142/S0218202517500282. Google Scholar
[43]	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. Nat. Acad. Sci., USA, 102 (2005), 2277–2282 doi: 10.1073/pnas.0406724102. Google Scholar
[44]	M. Winkler, Does a "volume-filling effect" always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24. doi: 10.1002/mma.1146. Google Scholar
[45]	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
[46]	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
[47]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020. Google Scholar
[48]	M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Rational Mech. Anal., 211 (2014), 455-487. doi: 10.1007/s00205-013-0678-9. Google Scholar
[49]	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
[50]	M. Winkler, Global existence and slow grow-up in a quasilinear Keller-Segel system with exponentially decaying diffusivity, Nonlinearity, 30 (2017), 735-764. doi: 10.1088/1361-6544/aa565b. Google Scholar
[51]	C. Yoon and Y. J. Kim, Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion, Acta Application Mathematics, 149 (2017), 101-123. doi: 10.1007/s10440-016-0089-7. Google Scholar

show all references

References:

[1]	N. Bellomo, A. Bellouquid, Y. S. Tao and M. Winkler, Towards 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
[2]	J. P. Bourguignon and H. Brezis, Remarks on Euler Equation, J. Functional Analysis, 15 (1974), 341-363. doi: 10.1016/0022-1236(74)90027-5. Google Scholar
[3]	M. Chae, K. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297. doi: 10.3934/dcds.2013.33.2271. Google Scholar
[4]	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. Google Scholar
[5]	A. Chertock, K. Fellner, A. Kurganov, A. Lorz and P. A. Markowich, Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: A high-resolution numerical approach, J. Fluid Mech., 694 (2012), 155-190. doi: 10.1017/jfm.2011.534. Google Scholar
[6]	S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237. doi: 10.1016/0025-5564(81)90055-9. Google Scholar
[7]	T. Ciéslak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851. doi: 10.1016/j.jde.2012.01.045. Google Scholar
[8]	T. Ciéslak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel and applications to volume filling models, J. Differential Equations, 258 (2015), 2080-2113. doi: 10.1016/j.jde.2014.12.004. Google Scholar
[9]	M. DiFrancesco, A. Lorz and P. A. Markowich, Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior, Discrete Contin. Dyn. Syst., 28 (2010), 1437-1453. doi: 10.3934/dcds.2010.28.1437. Google Scholar
[10]	R. J. 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. Google Scholar
[11]	R. J. Duan and Z. Xiang, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, Int. Math. Res. Not. IMRN, 7 (2014), 1833-1852. doi: 10.1093/imrn/rns270. Google Scholar
[12]	M. Eisenbach, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion. Chemotaxis, Imperial College Press, London, 2004.Google Scholar
[13]	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. Google Scholar
[14]	X. Fu, H. Tang, C. Liu, J. D. Huang, T. Hwa and P. Lenz, Stripe formation in bacterial system with density-suppressed motility, Phys. Rev. Lett., 108 (2012), 198102. doi: 10.1103/PhysRevLett.108.198102. Google Scholar
[15]	Y. Giga, Solutions for semilinear parabolic equations in L^p and regularity of weak solution of the Navier-Stokes system, J. Differential Equations, 61 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3. Google Scholar
[16]	Y. Giga and H. Sohr, Abstract L^p estimate for the Cauchy problem with application to the Navier-Stokes system in exterior domains, J. Funct. Anal., 102 (1991), 72-94. doi: 10.1016/0022-1236(91)90136-S. Google Scholar
[17]	M. Herrero and J. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683. Google Scholar
[18]	T. Hillen and K. Painter, A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. Google Scholar
[19]	T. Hillen and A. Potapov, The one-dimensional chemotaxis model: Global existence and asymptotic profile, Math. Methods Appl. Sci., 27 (2004), 1783-1801. doi: 10.1002/mma.569. Google Scholar
[20]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅰ, Jahresber. Deutsch. Math.-Verien., 105 (2003), 103-165. Google Scholar
[21]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅱ, Jahresber. Deutsch. Math.-Verien., 106 (2004), 51-69. Google Scholar
[22]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022. Google Scholar
[23]	W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6. Google Scholar
[24]	H. Y. Jin, Y. J. Kim and Z. A. Wang, Boundedness, stabilization and pattern formation driven by density-suppressed motility, SIAM J. Appl. Math., 2018, to appear.Google Scholar
[25]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. Google Scholar
[26]	R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398. doi: 10.1016/j.jmaa.2008.01.005. Google Scholar
[27]	O. Ladyzhenskaya, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, RI, 1968. Google Scholar
[28]	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. Google Scholar
[29]	A. Lorz, Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004. doi: 10.1142/S0218202510004507. Google Scholar
[30]	A. Lorz, A coupled Keller-Segel-Stokes model: Global existence for small initial data and blow-up delay, Commun. Math. Sci., 10 (2012), 555-574. doi: 10.4310/CMS.2012.v10.n2.a7. Google Scholar
[31]	T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. Google Scholar
[32]	T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. Google Scholar
[33]	K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. Google Scholar
[34]	K. Painter and J. A. Sherratt, Modeling the movement of interacting cell populations, J. Theor. Biol., 225 (2003), 327-339. doi: 10.1016/S0022-5193(03)00258-3. Google Scholar
[35]	M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178. doi: 10.1006/jdeq.1993.1045. Google Scholar
[36]	T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367. doi: 10.4310/MAA.2001.v8.n2.a9. Google Scholar
[37]	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. Google Scholar
[38]	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. Google Scholar
[39]	Y. S. Tao and M. Winkler, Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250. doi: 10.1137/15M1014115. Google Scholar
[40]	Y. S. 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. Google Scholar
[41]	Y. S. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2016), 2555-2573. doi: 10.1007/s00033-015-0541-y. Google Scholar
[42]	Y. Tao and M. Winkler, Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system, Math. Models Meth. Appl. Sci., 27 (2017), 1645-1683. doi: 10.1142/S0218202517500282. Google Scholar
[43]	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. Nat. Acad. Sci., USA, 102 (2005), 2277–2282 doi: 10.1073/pnas.0406724102. Google Scholar
[44]	M. Winkler, Does a "volume-filling effect" always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24. doi: 10.1002/mma.1146. Google Scholar
[45]	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
[46]	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
[47]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020. Google Scholar
[48]	M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Rational Mech. Anal., 211 (2014), 455-487. doi: 10.1007/s00205-013-0678-9. Google Scholar
[49]	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
[50]	M. Winkler, Global existence and slow grow-up in a quasilinear Keller-Segel system with exponentially decaying diffusivity, Nonlinearity, 30 (2017), 735-764. doi: 10.1088/1361-6544/aa565b. Google Scholar
[51]	C. Yoon and Y. J. Kim, Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion, Acta Application Mathematics, 149 (2017), 101-123. doi: 10.1007/s10440-016-0089-7. Google Scholar

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