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July  2020, 19(7): 3843-3883. doi: 10.3934/cpaa.2020170

Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

2. 

Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China

3. 

Institute of Artificial Intelligence, Huazhong University of Science and Technology, Wuhan 430074, China

* Corresponding author

Received  January 2020 Revised  February 2020 Published  April 2020

Fund Project: The authors were supported by NSFC grant 11971185

In this paper, we investigate the chemotaxis-fluid system with singular sensitivity and logistic source in bounded convex domain with smooth boundary. We present the global existence of very weak solutions under appropriate regularity assumptions on the initial data. Then, we show that system possesses a global bounded classical solution. Finally, we present a unique globally bounded classical solution for a fluid-free system. In addition, the asymptotic behavior of the solutions is studied, and our results generalize and improve some well-known results in the literature, and partially results are new.

Citation: Guoqiang Ren, Bin Liu. Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3843-3883. doi: 10.3934/cpaa.2020170
References:
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N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Meth. Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.

[2]

T. Black, Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system in 2D, J. Differ. Equ., 265 (2018), 2296-2339.  doi: 10.1016/j.jde.2018.04.035.

[3]

T. Black, Global very weak solutions to a chemotaxis-fluid system with nonlinear diffusion, SIAM J. Math. Anal., 50 (2018), 4087-4116.  doi: 10.1137/17M1159488.

[4]

T. Black, Global solvability of chemotaxis-fluid systems with nonlinear diffusion and matrix-valued sensitivities in three dimensions, Nonlinear Anal., 180 (2019), 129-153.  doi: 10.1016/j.na.2018.10.003.

[5]

T. BlackJ. Lankeit and M. Mizukami, Singular sensitivity in a Keller-Segel-fluid system, J. Evol. Equ., 18 (2018), 561-581.  doi: 10.1007/s00028-017-0411-5.

[6]

M. DingW. Wang and S. Zhou, Global existence of solutions to a fully parabolic chemotaxis system with singular sensitivity and logistic source, Nonlinear Anal. Real World Appl., 49 (2019), 286-311.  doi: 10.1016/j.nonrwa.2019.03.009.

[7]

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.

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D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.

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C. Jin, Global solvability and boundedness to a coupled chemotaxis-fluid model with arbitrary porous medium diffusion, J. Differ. Equ., 265 (2018), 332-353.  doi: 10.1016/j.jde.2018.02.031.

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C. Jin, Large time periodic solutions to coupled chemotaxis-fluid models, Z. Angew. Math. Phys., 68 (2017), 137. doi: 10.1007/s00033-017-0882-9.

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E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

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E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 27 (1971), 235-248. 

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S. Kurima and M. Mizukami, Global weak solutions to a 3-dimensional degenerate and singular chemotaxis-Navier-Stokes system with logistic source, Nonlinear Anal. Real World Appl., 46 (2019), 98-115.  doi: 10.1016/j.nonrwa.2018.09.011.

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E. Lankeit and J. Lankeit, On the global generalized solvability of a chemotaxis model with signal absorption and logistic growth terms, Nonlinearity, 32 (2019), 1569-1596.  doi: 10.1088/1361-6544/aaf8c0.

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B. Liu and G. Ren, Global existence and asymptotic behavior in a three-dimensional two-species chemotaxis-Stokes system with tensor-valued sensitivity, J. Korean Math. Soc., 57 (2020), 215-247.  doi: 10.4134/JKMS.j190028.

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J. Liu and Y. Wang, Boundedness and decay property in a three-dimensional Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation, J. Differ. Equ., 261 (2016), 967-999.  doi: 10.1016/j.jde.2016.03.030.

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G. Ren and B. Liu, Boundedness of solutions for a quasilinear chemotaxis-haptotaxis model, Hakkaido Math. J., (2019), Accepted. doi: 10.1002/mma.4126.

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G. Ren and B. Liu, Global boundedness and asymptotic behavior in a two-species chemotaxis-competition system with two signals, Nonlinear Anal. Real World Appl., 48 (2019), 288-325.  doi: 10.1016/j.nonrwa.2019.01.017.

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G. Ren and B. Liu, Global dynamics for an attraction-repulsion chemotaxis model with logistic source, J. Differ. Equ., 268 (2020), 4320-4373.  doi: 10.1016/j.jde.2019.10.027.

[24]

G. Ren and B. Liu, Global existence and asymptotic behavior in a two-species chemotaxis system with logistic source, J. Differ. Equ., 269 (2020), 1484-1520. doi: 10.1016/j.jde.2020.01.008.

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G. Ren and B. Liu, Global existence of bounded solutions for a quasilinear chemotaxis system with logistic source, Nonlinear Anal. Real World Appl., 46 (2019), 545-582.  doi: 10.1016/j.nonrwa.2018.09.020.

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Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. Henri Poincare Anal. Non Lineaire, 30 (2013), 157-178.  doi: 10.1016/j.anihpc.2012.07.002.

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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.

[29]

Y. Tao and M. Winkler, Large time behavior in a multi-dimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.  doi: 10.1137/15M1014115.

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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), 1-23.  doi: 10.1007/s00033-016-0732-1.

[31]

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.

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I. TuvalL. CisnerosC. DombrowskiC. W. WolgemuthJ. 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.

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G. Viglialoro, Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl., 439 (2016), 197-212.  doi: 10.1016/j.jmaa.2016.02.069.

[34]

G. Viglialoro, Boundedness properties of very weak solutions to a fully parabolic chemotaxis-system with logistic source, Nonlinear Anal. Real World Appl., 34 (2017), 520-535.  doi: 10.1016/j.nonrwa.2016.10.001.

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Y. Wang, Global large-data generalized solutions in a two-dimensional chemotaxis-Stokes system with singular sensitivity, Bound. Value Probl., 2016 (2016), 177. doi: 10.1186/s13661-016-0687-3.

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Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with subcritical sensitivity, Math. Models Meth. Appl. Sci., 27 (2017), 2745-2780.  doi: 10.1142/S0218202517500579.

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Y. WangM. Winkler and Z. Xiang, Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 18 (2018), 421-466.  doi: 10.1109/tps.2017.2783887.

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Y. Wang, M. Winkler and Z. Xiang, The small-convection limit in a two-dimensional chemotaxis-Navier-Stokes system, Math. Z., 289 (2018), 71-108. doi: 10.1007/s00209-017-1944-6.

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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. Differ. Equ., 261 (2016), 4944-4973.  doi: 10.1016/j.jde.2016.07.010.

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M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equ., 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.

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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.

[44]

M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differ. Equ., 54 (2015), 3789-3828.  doi: 10.1007/s00526-015-0922-2.

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show all references

References:
[1]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Meth. Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.

[2]

T. Black, Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system in 2D, J. Differ. Equ., 265 (2018), 2296-2339.  doi: 10.1016/j.jde.2018.04.035.

[3]

T. Black, Global very weak solutions to a chemotaxis-fluid system with nonlinear diffusion, SIAM J. Math. Anal., 50 (2018), 4087-4116.  doi: 10.1137/17M1159488.

[4]

T. Black, Global solvability of chemotaxis-fluid systems with nonlinear diffusion and matrix-valued sensitivities in three dimensions, Nonlinear Anal., 180 (2019), 129-153.  doi: 10.1016/j.na.2018.10.003.

[5]

T. BlackJ. Lankeit and M. Mizukami, Singular sensitivity in a Keller-Segel-fluid system, J. Evol. Equ., 18 (2018), 561-581.  doi: 10.1007/s00028-017-0411-5.

[6]

M. DingW. Wang and S. Zhou, Global existence of solutions to a fully parabolic chemotaxis system with singular sensitivity and logistic source, Nonlinear Anal. Real World Appl., 49 (2019), 286-311.  doi: 10.1016/j.nonrwa.2019.03.009.

[7]

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.

[8]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.

[9]

C. Jin, Global solvability and boundedness to a coupled chemotaxis-fluid model with arbitrary porous medium diffusion, J. Differ. Equ., 265 (2018), 332-353.  doi: 10.1016/j.jde.2018.02.031.

[10]

C. Jin, Large time periodic solutions to coupled chemotaxis-fluid models, Z. Angew. Math. Phys., 68 (2017), 137. doi: 10.1007/s00033-017-0882-9.

[11]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[12]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 27 (1971), 235-248. 

[13]

S. Kurima and M. Mizukami, Global weak solutions to a 3-dimensional degenerate and singular chemotaxis-Navier-Stokes system with logistic source, Nonlinear Anal. Real World Appl., 46 (2019), 98-115.  doi: 10.1016/j.nonrwa.2018.09.011.

[14]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equation of Parabolic Type, Amer. Math. Soc. Transl., vol. 23, American Mathematical Society, Providence, RI, 1968.

[15]

E. Lankeit and J. Lankeit, On the global generalized solvability of a chemotaxis model with signal absorption and logistic growth terms, Nonlinearity, 32 (2019), 1569-1596.  doi: 10.1088/1361-6544/aaf8c0.

[16]

B. Liu and G. Ren, Global existence and asymptotic behavior in a three-dimensional two-species chemotaxis-Stokes system with tensor-valued sensitivity, J. Korean Math. Soc., 57 (2020), 215-247.  doi: 10.4134/JKMS.j190028.

[17]

J. Liu and Y. Wang, Boundedness and decay property in a three-dimensional Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation, J. Differ. Equ., 261 (2016), 967-999.  doi: 10.1016/j.jde.2016.03.030.

[18]

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. Differ. Equ., 262 (2017), 5271-5305.  doi: 10.1016/j.jde.2017.01.024.

[19]

H. Matthias and P. Jan, Heat kernels and maximal $L^p$-$L^q$ estimates for parabolic evolution equations, Commun. Partial Differ. Equ., 22 (1997), 1647-1669.  doi: 10.1080/03605309708821314.

[20]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. 

[21]

G. Ren and B. Liu, Boundedness of solutions for a quasilinear chemotaxis-haptotaxis model, Hakkaido Math. J., (2019), Accepted. doi: 10.1002/mma.4126.

[22]

G. Ren and B. Liu, Global boundedness and asymptotic behavior in a two-species chemotaxis-competition system with two signals, Nonlinear Anal. Real World Appl., 48 (2019), 288-325.  doi: 10.1016/j.nonrwa.2019.01.017.

[23]

G. Ren and B. Liu, Global dynamics for an attraction-repulsion chemotaxis model with logistic source, J. Differ. Equ., 268 (2020), 4320-4373.  doi: 10.1016/j.jde.2019.10.027.

[24]

G. Ren and B. Liu, Global existence and asymptotic behavior in a two-species chemotaxis system with logistic source, J. Differ. Equ., 269 (2020), 1484-1520. doi: 10.1016/j.jde.2020.01.008.

[25]

G. Ren and B. Liu, Global existence of bounded solutions for a quasilinear chemotaxis system with logistic source, Nonlinear Anal. Real World Appl., 46 (2019), 545-582.  doi: 10.1016/j.nonrwa.2018.09.020.

[26]

H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8255-2.

[27]

Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. Henri Poincare Anal. Non Lineaire, 30 (2013), 157-178.  doi: 10.1016/j.anihpc.2012.07.002.

[28]

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.

[29]

Y. Tao and M. Winkler, Large time behavior in a multi-dimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.  doi: 10.1137/15M1014115.

[30]

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), 1-23.  doi: 10.1007/s00033-016-0732-1.

[31]

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.

[32]

I. TuvalL. CisnerosC. DombrowskiC. W. WolgemuthJ. 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.

[33]

G. Viglialoro, Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl., 439 (2016), 197-212.  doi: 10.1016/j.jmaa.2016.02.069.

[34]

G. Viglialoro, Boundedness properties of very weak solutions to a fully parabolic chemotaxis-system with logistic source, Nonlinear Anal. Real World Appl., 34 (2017), 520-535.  doi: 10.1016/j.nonrwa.2016.10.001.

[35]

Y. Wang, Global large-data generalized solutions in a two-dimensional chemotaxis-Stokes system with singular sensitivity, Bound. Value Probl., 2016 (2016), 177. doi: 10.1186/s13661-016-0687-3.

[36]

Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with subcritical sensitivity, Math. Models Meth. Appl. Sci., 27 (2017), 2745-2780.  doi: 10.1142/S0218202517500579.

[37]

Y. WangM. Winkler and Z. Xiang, Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 18 (2018), 421-466.  doi: 10.1109/tps.2017.2783887.

[38]

Y. Wang, M. Winkler and Z. Xiang, The small-convection limit in a two-dimensional chemotaxis-Navier-Stokes system, Math. Z., 289 (2018), 71-108. doi: 10.1007/s00209-017-1944-6.

[39]

Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation, J. Differ. Equ., 259 (2015), 7578-7609.  doi: 10.1016/j.jde.2015.08.027.

[40]

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. Differ. Equ., 261 (2016), 4944-4973.  doi: 10.1016/j.jde.2016.07.010.

[41]

M. Winkler, Chemotaxis with logistic source: very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.  doi: 10.1016/j.jmaa.2008.07.071.

[42]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equ., 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.

[43]

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.

[44]

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