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February  2019, 24(2): 907-929. doi: 10.3934/dcdsb.2018212

Global weak solutions for a coupled chemotaxis non-Newtonian fluid

Hafedh Bousbih ,

University of Sousse, Higher Institute of Applied Sciences and Technology of Sousse, Ibn Khaldoun District, Sousse 4003, Tunisia

* Corresponding author: hafedh.bousbih@gmail.com

Received  October 2017 Revised  March 2018 Published  June 2018

This paper focuses on the mathematical analysis of a self-suggested model arising from biology, consisting of dynamics of oxygen diffusion and consumption, chemotaxis process and viscous incompressible non-Newtonian fluid in a bounded domain $Ω \subset \mathbb{R}^d$, with $d = 2, 3.$ The viscosity of the studied fluid is supposed to be non constant and depends on the shear-rate $|{\bf{D}}\boldsymbol{v}|$ as well as the cell density $m$ and the oxygen concentration $c$. Nonlinearities are also considered in the diffusion terms for the convection-diffusion equations corresponding to $m$ and $c$. Under the choice of suitable structures and convenient assumptions for the nonlinear fluxes, we prove global existence of weak solutions, in the case of a smooth bounded domain subject to Navier's slip conditions at the boundary and for large range of initial data.

Keywords: Chemotaxis, non-Newtonian fluid, generalized Navier-Stokes system, reaction-diffusion equations, nonlinear diffusion, global existence, Navier's slip boundary conditions.
Mathematics Subject Classification: Primary: 35K57, 76D03, 76D05, 74F25, 92C17; Secondary: 35K55, 35Q92.
Citation: Hafedh Bousbih. Global weak solutions for a coupled chemotaxis non-Newtonian fluid. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 907-929. doi: 10.3934/dcdsb.2018212
References:
[1]

N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Mod. Meth. Appl. S., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X. Google Scholar

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J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin Heidelberg, 1976. Google Scholar

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[4]

M. Braukhoff, Global (weak) solution of the chemotaxis-Navier-Stokes equations with non-homogeneous boundary conditions and logistic growth, Ann. Inst. H. Poincaré (C) Non Linear Anal., 34 (2017), 1013-1039. doi: 10.1016/j.anihpc.2016.08.003. Google Scholar

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M. Bulíček, E. Feireisl and J. Málek, A Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients, Nonlinear Anal., 10 (2009), 992-1015. doi: 10.1016/j.nonrwa.2007.11.018. Google Scholar

[6]

M. Bulíček, J. Málek and K. R. Rajagopal, Navier's slip and evolutionary Navier Stokes like systems with pressure and shear-rate dependent viscosity, Indiana Univ. Math. J., 56 (2007), 51-85. doi: 10.1512/iumj.2007.56.2997. Google Scholar

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M. Bulíček, J. Málek and K.R. Rajagopal, Mathematical results concerning unsteady flows of chemically reacting incompressible fluids, in Partial Differential Equations and Fluid Mechanics, 364 (2009), 26-53. Google Scholar

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M. A. J. Chaplain, M. Ganesh and I. G. Graham, Spatio-temporal pattern formation on spherical surfaces: Numerical simulation and application to solid tumour growth, J. Math. Biol., 42 (2001), 387-423. doi: 10.1007/s002850000067. Google Scholar

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T. Clopeau, A. Mikelić and R. Robert, On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions, Nonlinearity, 11 (1998), 1625-1636. doi: 10.1088/0951-7715/11/6/011. Google Scholar

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C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein and J. O. Kessler, Self-concentration and large-scale coherence in bacterial dynamics, Phys. Rev. Lett., 93 (2004), 098103. doi: 10.1103/PhysRevLett.93.098103. Google Scholar

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R. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Part. Diff. Eq., 35 (2010), 1635-1673. doi: 10.1080/03605302.2010.497199. Google Scholar

[13]

R. Duan and Z. Xiang, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, Int. Math. Res. Notices, 7 (2014), 1833-1852. doi: 10.1093/imrn/rns270. Google Scholar

[14]

M. Eisenbach, Chemotaxis, Imperial College Press, London, 2004.Google Scholar

[15]

J. Frehse, J. Málek and M. Steinhauer, An existence result for fluids with shear dependent viscosity-steady flows, Nonlinear Anal, 30 (1997), 3041-3049. doi: 10.1016/S0362-546X(97)00392-1. Google Scholar

[16]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-state Problems, Springer-Verlag, New York, 2011. doi: 10.1007/978-0-387-09620-9. Google Scholar

[17]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. Google Scholar

[18]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. Google Scholar

[19]

E. Keller and L. 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. Google Scholar

[20]

E. Keller and L. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. Google Scholar

[21]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969. Google Scholar

[22]

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

[23]

A. Lorz, Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004. doi: 10.1142/S0218202510004507. Google Scholar

[24]

J. Málek, J. Nečas and K. R. Rajagopal, Global analysis of the flows of fluids with shear and pressure dependent viscosities, Arch. Ration. Mech. Anal., 165 (2002), 243-269. doi: 10.1007/s00205-002-0219-4. Google Scholar

[25]

J. Málek, J. Nečas, M. Rokyta and M. R${\rm{\dot u}}$žička, Weak and Measure-Valued Solutions to Evolutionnary PDE's, Chapman & Hall, London, 1996. doi: 10.1007/978-1-4899-6824-1. Google Scholar

[26]

J. Málek and K. R. Rajagopal, Mathematical issues concerning the Navier-Stokes equations and some of its generalizations, in Handb. Differ. Equ.: Evol. Equ., 2 (2005), 371-459. doi: 10.1016/S1874-5717(06)80008-3. Google Scholar

[27]

H. G. Othmer and T. Hillen, The Diffusion Limit of Transport Equations II: Chemotaxis Equations, SIAM J. Appl. Math., 62 (2002), 1222-1250. doi: 10.1137/S0036139900382772. Google Scholar

[28]

K. J. Painter, P. K. Maini and H. G. Othmer, Development and applications of a model for cellular response to multiple chemotactic cues, J. Math. Biol., 41 (2000), 285-314. doi: 10.1007/s002850000035. Google Scholar

[29]

K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Quart., 10 (2002), 501-543. Google Scholar

[30]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407. Google Scholar

[31]

Y. Peng and Z. Xiang, Global solutions to the coupled chemotaxis-fluids system in a 3D unbounded domain with boundary, Math. Models Methods Appl. Sci., 28 (2018), 455-487. doi: 10.1142/S0218202518500239. Google Scholar

[32]

T. Roubícěk, A generalization of the Lions-Temam compact imbedding theorem, Časopis Pěst. Mat., 115 (1990), 338–342. https://www.researchgate.net/publication/265639524_A_generalization_of_the_Lions-Temam_compact_imbedding_theorem Google Scholar

[33]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl.(4), 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar

[34]

Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 1901-1914. doi: 10.3934/dcds.2012.32.1901. Google Scholar

[35]

Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with non-linear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178. doi: 10.1016/j.anihpc.2012.07.002. Google Scholar

[36]

I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen tansport near contact lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277-2282. Google Scholar

[37]

D. Vorotnikov, Weak solutions for a bioconvection model related to Bacillus subtilis, Commun. Math. Sci., 12 (2014), 545-563. doi: 10.4310/CMS.2014.v12.n3.a8. Google Scholar

[38]

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

[39]

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. Google Scholar

[40]

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. Google Scholar

[41]

E. Zeidler, Nonlinear Functional Analysis and Its Applications, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5. Google Scholar

[42]

Q. Zhang and Y. Li, Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 259 (2015), 3730-3754. doi: 10.1016/j.jde.2015.05.012. Google Scholar

show all references

References:
[1]

N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Mod. Meth. Appl. S., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X. Google Scholar

[2]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin Heidelberg, 1976. Google Scholar

[3]

L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., 19 (1992), 581-597. doi: 10.1016/0362-546X(92)90023-8. Google Scholar

[4]

M. Braukhoff, Global (weak) solution of the chemotaxis-Navier-Stokes equations with non-homogeneous boundary conditions and logistic growth, Ann. Inst. H. Poincaré (C) Non Linear Anal., 34 (2017), 1013-1039. doi: 10.1016/j.anihpc.2016.08.003. Google Scholar

[5]

M. Bulíček, E. Feireisl and J. Málek, A Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients, Nonlinear Anal., 10 (2009), 992-1015. doi: 10.1016/j.nonrwa.2007.11.018. Google Scholar

[6]

M. Bulíček, J. Málek and K. R. Rajagopal, Navier's slip and evolutionary Navier Stokes like systems with pressure and shear-rate dependent viscosity, Indiana Univ. Math. J., 56 (2007), 51-85. doi: 10.1512/iumj.2007.56.2997. Google Scholar

[7]

M. Bulíček, J. Málek and K.R. Rajagopal, Mathematical results concerning unsteady flows of chemically reacting incompressible fluids, in Partial Differential Equations and Fluid Mechanics, 364 (2009), 26-53. Google Scholar

[8]

M. A. J. Chaplain, M. Ganesh and I. G. Graham, Spatio-temporal pattern formation on spherical surfaces: Numerical simulation and application to solid tumour growth, J. Math. Biol., 42 (2001), 387-423. doi: 10.1007/s002850000067. Google Scholar

[9]

T. Clopeau, A. Mikelić and R. Robert, On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions, Nonlinearity, 11 (1998), 1625-1636. doi: 10.1088/0951-7715/11/6/011. Google Scholar

[10]

M. Di Francesco, A. Lorz and P. Markowich, Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior, Discrete Contin. Dyn. Syst. Ser. A, 28 (2010), 1437-1453. doi: 10.3934/dcds.2010.28.1437. Google Scholar

[11]

C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein and J. O. Kessler, Self-concentration and large-scale coherence in bacterial dynamics, Phys. Rev. Lett., 93 (2004), 098103. doi: 10.1103/PhysRevLett.93.098103. Google Scholar

[12]

R. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Part. Diff. Eq., 35 (2010), 1635-1673. doi: 10.1080/03605302.2010.497199. Google Scholar

[13]

R. Duan and Z. Xiang, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, Int. Math. Res. Notices, 7 (2014), 1833-1852. doi: 10.1093/imrn/rns270. Google Scholar

[14]

M. Eisenbach, Chemotaxis, Imperial College Press, London, 2004.Google Scholar

[15]

J. Frehse, J. Málek and M. Steinhauer, An existence result for fluids with shear dependent viscosity-steady flows, Nonlinear Anal, 30 (1997), 3041-3049. doi: 10.1016/S0362-546X(97)00392-1. Google Scholar

[16]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-state Problems, Springer-Verlag, New York, 2011. doi: 10.1007/978-0-387-09620-9. Google Scholar

[17]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. Google Scholar

[18]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. Google Scholar

[19]

E. Keller and L. 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. Google Scholar

[20]

E. Keller and L. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. Google Scholar

[21]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969. Google Scholar

[22]

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

[23]

A. Lorz, Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004. doi: 10.1142/S0218202510004507. Google Scholar

[24]

J. Málek, J. Nečas and K. R. Rajagopal, Global analysis of the flows of fluids with shear and pressure dependent viscosities, Arch. Ration. Mech. Anal., 165 (2002), 243-269. doi: 10.1007/s00205-002-0219-4. Google Scholar

[25]

J. Málek, J. Nečas, M. Rokyta and M. R${\rm{\dot u}}$žička, Weak and Measure-Valued Solutions to Evolutionnary PDE's, Chapman & Hall, London, 1996. doi: 10.1007/978-1-4899-6824-1. Google Scholar

[26]

J. Málek and K. R. Rajagopal, Mathematical issues concerning the Navier-Stokes equations and some of its generalizations, in Handb. Differ. Equ.: Evol. Equ., 2 (2005), 371-459. doi: 10.1016/S1874-5717(06)80008-3. Google Scholar

[27]

H. G. Othmer and T. Hillen, The Diffusion Limit of Transport Equations II: Chemotaxis Equations, SIAM J. Appl. Math., 62 (2002), 1222-1250. doi: 10.1137/S0036139900382772. Google Scholar

[28]

K. J. Painter, P. K. Maini and H. G. Othmer, Development and applications of a model for cellular response to multiple chemotactic cues, J. Math. Biol., 41 (2000), 285-314. doi: 10.1007/s002850000035. Google Scholar

[29]

K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Quart., 10 (2002), 501-543. Google Scholar

[30]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407. Google Scholar

[31]

Y. Peng and Z. Xiang, Global solutions to the coupled chemotaxis-fluids system in a 3D unbounded domain with boundary, Math. Models Methods Appl. Sci., 28 (2018), 455-487. doi: 10.1142/S0218202518500239. Google Scholar

[32]

T. Roubícěk, A generalization of the Lions-Temam compact imbedding theorem, Časopis Pěst. Mat., 115 (1990), 338–342. https://www.researchgate.net/publication/265639524_A_generalization_of_the_Lions-Temam_compact_imbedding_theorem Google Scholar

[33]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl.(4), 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar

[34]

Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 1901-1914. doi: 10.3934/dcds.2012.32.1901. Google Scholar

[35]

Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with non-linear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178. doi: 10.1016/j.anihpc.2012.07.002. Google Scholar

[36]

I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen tansport near contact lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277-2282. Google Scholar

[37]

D. Vorotnikov, Weak solutions for a bioconvection model related to Bacillus subtilis, Commun. Math. Sci., 12 (2014), 545-563. doi: 10.4310/CMS.2014.v12.n3.a8. Google Scholar

[38]

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

[39]

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. Google Scholar

[40]

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. Google Scholar

[41]

E. Zeidler, Nonlinear Functional Analysis and Its Applications, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5. Google Scholar

[42]

Q. Zhang and Y. Li, Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 259 (2015), 3730-3754. doi: 10.1016/j.jde.2015.05.012. Google Scholar

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