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.
Citation: |
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 Lp(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.![]() ![]() ![]() |