Global weak solutions for a coupled chemotaxis non-Newtonian fluid

Hafedh Bousbih

doi:10.3934/dcdsb.2018212

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2019, Volume 24, Issue 2: 907-929. Doi: 10.3934/dcdsb.2018212

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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
* Corresponding author: hafedh.bousbih@gmail.com

Received: September 30, 2017

Revised: February 28, 2018

Early access: June 2018

Published: January 2019

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K57, 76D03, 76D05, 74F25, 92C17; Secondary: 35K55, 35Q92.

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References

	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.
	J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin Heidelberg, 1976.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	M. Eisenbach, Chemotaxis, Imperial College Press, London, 2004.
	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.
	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.
	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.
	D. Horstmann , From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003) , 103-165.
	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.
	E. Keller and L. Segel , Model for chemotaxis, J. Theor. Biol., 30 (1971) , 225-234. doi: 10.1016/0022-5193(71)90050-6.
	O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.
	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.
	A. Lorz , Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010) , 987-1004. doi: 10.1142/S0218202510004507.
	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.
	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.
	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.
	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.
	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.
	K. Painter and T. Hillen , Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Quart., 10 (2002) , 501-543.
	C. S. Patlak , Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953) , 311-338. doi: 10.1007/BF02476407.
	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.
	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
	J. Simon , Compact sets in the space L^p(0, T; B), Ann. Mat. Pura Appl.(4), 146 (1987) , 65-96. doi: 10.1007/BF01762360.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	E. Zeidler, Nonlinear Functional Analysis and Its Applications, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.
	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.