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Existence and uniqueness of solutions of free boundary problems in heterogeneous environments
Global existence for an attraction-repulsion chemotaxis fluid model with logistic source
1. | Universidad Industrial de Santander, Escuela de Matemáticas, Bucaramanga, A.A. 678, Colombia |
2. | Universidade Estadual de Campinas, Departamento de Matemática-IMECC, CEP 13083-859, Campinas-SP, Brazil |
We consider an attraction-repulsion chemotaxis model coupled with the Navier-Stokes system. This model describes the interaction between a type of cells (e.g., bacteria), which proliferate following a logistic law, and two chemical signals produced by the cells themselves that degraded at a constant rate. Also, it is considered that the chemoattractant is consumed with a rate proportional to the amount of organisms. The cells and chemical substances are transported by a viscous incompressible fluid under the influence of a force due to the aggregation of cells. We prove the existence of global mild solutions in bounded domains of $\mathbb{R}^N,$ $N = 2, 3,$ for small initial data in $L^p$-spaces.
References:
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M. Aida, K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Chemotaxis and growth system with singular sensitivity function, Nonlinear Analysis: Real World Applications, 6 (2005), 323-336.
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[2] |
L. Angiuli, D. Pallara and F. Y. Paronetto,
Analytic semigroups generated in L1 by second order elliptic operators via duality methods, Semigroup Forum, Springer, 80 (2010), 255-271.
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[3] |
N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler,
Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Mathematical Models and Methods in Applied Sciences, 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[4] |
M. Braukhoff,
Global (weak) solution of the chemotaxis-Navier-Stokes equations with non-homogeneous boundary conditions and logistic growth, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 34 (2017), 1013-1039.
doi: 10.1016/j.anihpc.2016.08.003. |
[5] |
X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities,
Calc. Var. Partial Differential Equations, 55 (2016), Art. 107, 39 pp.
doi: 10.1007/s00526-016-1027-2. |
[6] |
T. Cazenave and F. B. Weissler,
Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations, Mathematische Zeitschrift, 228 (1998), 83-120.
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[7] |
S. Chandrasekhar,
Hydrodynamic and Hydromagnetic Stability, The International Series of Monographs on Physics Clarendon Press, Oxford, 1961. |
[8] |
M.A. Chaplain and G. Lolas,
Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Mathematical Models and Methods in Applied Sciences, 15 (2005), 1685-1734.
doi: 10.1142/S0218202505000947. |
[9] |
M. A. Chaplain and A. Stuart,
A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor, Mathematical Medicine and Biology, 10 (1993), 149-168.
doi: 10.1093/imammb/10.3.149. |
[10] |
A. Chertock, K. Fellner, A. Kurganov, A. Lorz and P. Markowich,
Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach, Journal of Fluid Mechanics, 694 (2012), 155-190.
doi: 10.1017/jfm.2011.534. |
[11] |
H. J. Choe and B. Lkhagvasuren,
Global existence result for chemotaxis Navier-Stokes equations in the critical Besov spaces, Journal of Mathematical Analysis and Applications, 446 (2017), 1415-1426.
doi: 10.1016/j.jmaa.2016.09.050. |
[12] |
C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein and J. O. Kessler,
Self-concentration and large-scale coherence in bacterial dynamics, Physical Review Letters, 93 (2004), 98-103.
|
[13] |
R. Duan and Z. Xiang,
A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, International Mathematics Research Notices, 2014 (2012), 1833-1852.
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[14] |
E. Espejo and T. Suzuki,
Reaction terms avoiding aggregation in slow fluids, Nonlinear Analysis: Real World Applications, 21 (2015), 110-126.
doi: 10.1016/j.nonrwa.2014.07.001. |
[15] |
L. C. F. Ferreira and E. J. Villamizar-Roa,
Self-similar solutions, uniqueness and long-time asymptotic behavior for semilinear heat equations, Differential and Integral Equations, 19 (2006), 1349-1370.
|
[16] |
L. C. F. Ferreira and E. J. Villamizar-Roa,
Well-posedness and asymptotic behaviour for the convection problem, Nonlinearity, 19 (2006), 2169-2191.
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D. Fujiwara and H. Morimoto,
An Lr-theorem of the Helmholtz decomposition of vector fields, IA Math, 24 (1977), 685-700.
|
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Y. Giga,
Analyticity of the semigroup generated by the Stokes operator in Lr spaces, Mathematische Zeitschrift, 178 (1981), 297-329.
doi: 10.1007/BF01214869. |
[19] |
N. A. Hill and T. J. Pedley,
Bioconvection, Fluid Dynamics Research, 37 (2005), 1-20.
doi: 10.1016/j.fluiddyn.2005.03.002. |
[20] |
T. Hillen and K. J. Painter,
A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[21] |
T. Hillen, K. J. Painter and M. Winkler,
Convergence of a cancer invasion model to a logistic chemotaxis model, Mathematical Models and Methods in Applied Sciences, 23 (2013), 165-198.
doi: 10.1142/S0218202512500480. |
[22] |
D. Horstmann,
Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, Journal of Nonlinear Science, 21 (2011), 231-270.
doi: 10.1007/s00332-010-9082-x. |
[23] |
S. Ishida,
Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains, Discrete & Continuous Dynamical Systems-A, 35 (2015), 3463-3482.
doi: 10.3934/dcds.2015.35.3463. |
[24] |
J. Jiang, H. Wu and S. Zheng,
Global existence and asymptotic behavior of solutions to a chemotaxis-fluid system on general bounded domains, Asymptotic Analysis, 92 (2015), 249-258.
|
[25] |
T. Kato,
Strong Lp-solutions of the Navier-Stokes equation in Rm with applications to weak solutions, Math. Z., 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[26] |
T. Kato,
Strong solutions of the Navier-Stokes equation in Morrey spaces, Bol. Soc. Brasil. Mat., 22 (1992), 127-155.
doi: 10.1007/BF01232939. |
[27] |
A. Kiselev and L. Ryzhik,
Biomixing by chemotaxis and enhancement of biological reactions, Communications in Partial Differential Equations, 37 (2012), 298-318.
doi: 10.1080/03605302.2011.589879. |
[28] |
H. Kozono, M. Miura and Y. Sugiyama,
Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid, Journal of Functional Analysis, 270 (2016), 1663-1683.
doi: 10.1016/j.jfa.2015.10.016. |
[29] |
J. Lankeit,
Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, Journal of Differential Equations, 258 (2015), 1158-1191.
doi: 10.1016/j.jde.2014.10.016. |
[30] |
J. Lankeit,
Long-term behaviour in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci., 26 (2016), 2071-2109.
doi: 10.1142/S021820251640008X. |
[31] |
D. Li, C. Mu, K. Lin and L. Wang,
Large time behavior of solution to an attraction-repulsion chemotaxis system with logistic source in three dimensions, Journal of Mathematical Analysis and Applications, 448 (2016), 914-936.
doi: 10.1016/j.jmaa.2016.11.036. |
[32] |
X. Li,
Boundedness in a two-dimensional attraction-repulsion system with nonlinear diffusion, Mathematical Methods in the Applied Sciences, 39 (2016), 289-301.
doi: 10.1002/mma.3477. |
[33] |
X. Li and Z. Xiang,
On an attraction-repulsion chemotaxis system with a logistic source, IMA Journal of Applied Mathematics, 81 (2016), 165-198.
doi: 10.1093/imamat/hxv033. |
[34] |
J. Liu and Y. Wang,
Global existence and boundedness in a Keller-Segel-(Navier-) Stokes system with signal-dependent sensitivity, Journal of Mathematical Analysis and Applications, 447 (2017), 499-528.
doi: 10.1016/j.jmaa.2016.10.028. |
[35] |
J. Liu and Y. Wang,
Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system involving a tensor-valued sensitivity with saturation, Journal of Differential Equations, 262 (2017), 5271-5305.
doi: 10.1016/j.jde.2017.01.024. |
[36] |
P. Liu, J. Shi and Z.-A. Wang,
Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625.
doi: 10.3934/dcdsb.2013.18.2597. |
[37] |
M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner,
Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: Is there a connection?, Bulletin of Mathematical Biology, 65 (2003), 693-730.
doi: 10.1016/S0092-8240(03)00030-2. |
[38] |
A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser/Springer Basel AG, Basel, 1995. |
[39] |
N. V. Mantzaris, S. Webb and H. G. Othmer,
Mathematical modeling of tumor-induced angiogenesis, Journal of Mathematical Biology, 49 (2004), 111-187.
doi: 10.1007/s00285-003-0262-2. |
[40] |
X. Mora,
Semilinear parabolic problems define semiflows on Ck spaces, Transactions of the American Mathematical Society, 278 (1983), 21-55.
doi: 10.2307/1999300. |
[41] |
A. Quinlan and B. Straughan,
Decay bounds in a model for aggregation of microglia: Application to Alzheimer's disease senile plaques, Proceedings of the Royal Society A, 461 (2005), 2887-2897.
doi: 10.1098/rspa.2005.1483. |
[42] |
Y. Tao and M. Winkler,
Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 30 (2013), 157-178.
doi: 10.1016/j.anihpc.2012.07.002. |
[43] |
Y. Tao and M. Winkler,
Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Zeitschrift für angewandte Mathematik und Physik, 66 (2015), 2555-2573.
doi: 10.1007/s00033-015-0541-y. |
[44] |
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), Art. 138, 23 pp.
doi: 10.1007/s00033-016-0732-1. |
[45] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Communications in Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[46] |
I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein,
Bacterial swimming and oxygen transport near contact lines, Proceedings of the National Academy of Sciences of the United States of America, 102 (2005), 2277-2282.
doi: 10.1073/pnas.0406724102. |
[47] |
R. Tyson, S. R. Lubkin and J. D. Murray,
Model and analysis of chemotactic bacterial patterns in a liquid medium, Journal of Mathematical Biology, 38 (1999), 359-375.
doi: 10.1007/s002850050153. |
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Y. Wang,
Boundedness in a three-dimensional attraction-repulsion chemotaxis system with nonlinear diffusion and logistic source, Electronic Journal of Differential Equations, 176 (2016), 1-21.
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Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation: The 3D case, Journal of Differential Equations, 261 (2016), 4944-4973.
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Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905.
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M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Communications in Partial Differential Equations, 35 (2010), 1516-1537.
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M. Winkler,
Global large-data solutions in a chemotaxis-(Navier-) Stokes system modeling cellular swimming in fluid drops, Communications in Partial Differential Equations, 37 (2012), 319-351.
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M. Winkler,
Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, Journal of Differential Equations, 257 (2014), 1056-1077.
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M. Winkler,
Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calculus of Variations and Partial Differential Equations, 54 (2015), 3789-3828.
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M. Winkler,
Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 33 (2016), 1329-1352.
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An attraction-repulsion chemotaxis system with logistic source, Biophysical Journal, 96 (2015), 570-584.
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Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion, Journal of Differential Equations, 259 (2015), 3730-3754.
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show all references
References:
[1] |
M. Aida, K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Chemotaxis and growth system with singular sensitivity function, Nonlinear Analysis: Real World Applications, 6 (2005), 323-336.
doi: 10.1016/j.nonrwa.2004.08.011. |
[2] |
L. Angiuli, D. Pallara and F. Y. Paronetto,
Analytic semigroups generated in L1 by second order elliptic operators via duality methods, Semigroup Forum, Springer, 80 (2010), 255-271.
doi: 10.1007/s00233-009-9200-y. |
[3] |
N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler,
Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Mathematical Models and Methods in Applied Sciences, 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[4] |
M. Braukhoff,
Global (weak) solution of the chemotaxis-Navier-Stokes equations with non-homogeneous boundary conditions and logistic growth, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 34 (2017), 1013-1039.
doi: 10.1016/j.anihpc.2016.08.003. |
[5] |
X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities,
Calc. Var. Partial Differential Equations, 55 (2016), Art. 107, 39 pp.
doi: 10.1007/s00526-016-1027-2. |
[6] |
T. Cazenave and F. B. Weissler,
Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations, Mathematische Zeitschrift, 228 (1998), 83-120.
doi: 10.1007/PL00004606. |
[7] |
S. Chandrasekhar,
Hydrodynamic and Hydromagnetic Stability, The International Series of Monographs on Physics Clarendon Press, Oxford, 1961. |
[8] |
M.A. Chaplain and G. Lolas,
Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Mathematical Models and Methods in Applied Sciences, 15 (2005), 1685-1734.
doi: 10.1142/S0218202505000947. |
[9] |
M. A. Chaplain and A. Stuart,
A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor, Mathematical Medicine and Biology, 10 (1993), 149-168.
doi: 10.1093/imammb/10.3.149. |
[10] |
A. Chertock, K. Fellner, A. Kurganov, A. Lorz and P. Markowich,
Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach, Journal of Fluid Mechanics, 694 (2012), 155-190.
doi: 10.1017/jfm.2011.534. |
[11] |
H. J. Choe and B. Lkhagvasuren,
Global existence result for chemotaxis Navier-Stokes equations in the critical Besov spaces, Journal of Mathematical Analysis and Applications, 446 (2017), 1415-1426.
doi: 10.1016/j.jmaa.2016.09.050. |
[12] |
C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein and J. O. Kessler,
Self-concentration and large-scale coherence in bacterial dynamics, Physical Review Letters, 93 (2004), 98-103.
|
[13] |
R. Duan and Z. Xiang,
A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, International Mathematics Research Notices, 2014 (2012), 1833-1852.
doi: 10.1093/imrn/rns270. |
[14] |
E. Espejo and T. Suzuki,
Reaction terms avoiding aggregation in slow fluids, Nonlinear Analysis: Real World Applications, 21 (2015), 110-126.
doi: 10.1016/j.nonrwa.2014.07.001. |
[15] |
L. C. F. Ferreira and E. J. Villamizar-Roa,
Self-similar solutions, uniqueness and long-time asymptotic behavior for semilinear heat equations, Differential and Integral Equations, 19 (2006), 1349-1370.
|
[16] |
L. C. F. Ferreira and E. J. Villamizar-Roa,
Well-posedness and asymptotic behaviour for the convection problem, Nonlinearity, 19 (2006), 2169-2191.
doi: 10.1088/0951-7715/19/9/011. |
[17] |
D. Fujiwara and H. Morimoto,
An Lr-theorem of the Helmholtz decomposition of vector fields, IA Math, 24 (1977), 685-700.
|
[18] |
Y. Giga,
Analyticity of the semigroup generated by the Stokes operator in Lr spaces, Mathematische Zeitschrift, 178 (1981), 297-329.
doi: 10.1007/BF01214869. |
[19] |
N. A. Hill and T. J. Pedley,
Bioconvection, Fluid Dynamics Research, 37 (2005), 1-20.
doi: 10.1016/j.fluiddyn.2005.03.002. |
[20] |
T. Hillen and K. J. Painter,
A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[21] |
T. Hillen, K. J. Painter and M. Winkler,
Convergence of a cancer invasion model to a logistic chemotaxis model, Mathematical Models and Methods in Applied Sciences, 23 (2013), 165-198.
doi: 10.1142/S0218202512500480. |
[22] |
D. Horstmann,
Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, Journal of Nonlinear Science, 21 (2011), 231-270.
doi: 10.1007/s00332-010-9082-x. |
[23] |
S. Ishida,
Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains, Discrete & Continuous Dynamical Systems-A, 35 (2015), 3463-3482.
doi: 10.3934/dcds.2015.35.3463. |
[24] |
J. Jiang, H. Wu and S. Zheng,
Global existence and asymptotic behavior of solutions to a chemotaxis-fluid system on general bounded domains, Asymptotic Analysis, 92 (2015), 249-258.
|
[25] |
T. Kato,
Strong Lp-solutions of the Navier-Stokes equation in Rm with applications to weak solutions, Math. Z., 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[26] |
T. Kato,
Strong solutions of the Navier-Stokes equation in Morrey spaces, Bol. Soc. Brasil. Mat., 22 (1992), 127-155.
doi: 10.1007/BF01232939. |
[27] |
A. Kiselev and L. Ryzhik,
Biomixing by chemotaxis and enhancement of biological reactions, Communications in Partial Differential Equations, 37 (2012), 298-318.
doi: 10.1080/03605302.2011.589879. |
[28] |
H. Kozono, M. Miura and Y. Sugiyama,
Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid, Journal of Functional Analysis, 270 (2016), 1663-1683.
doi: 10.1016/j.jfa.2015.10.016. |
[29] |
J. Lankeit,
Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, Journal of Differential Equations, 258 (2015), 1158-1191.
doi: 10.1016/j.jde.2014.10.016. |
[30] |
J. Lankeit,
Long-term behaviour in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci., 26 (2016), 2071-2109.
doi: 10.1142/S021820251640008X. |
[31] |
D. Li, C. Mu, K. Lin and L. Wang,
Large time behavior of solution to an attraction-repulsion chemotaxis system with logistic source in three dimensions, Journal of Mathematical Analysis and Applications, 448 (2016), 914-936.
doi: 10.1016/j.jmaa.2016.11.036. |
[32] |
X. Li,
Boundedness in a two-dimensional attraction-repulsion system with nonlinear diffusion, Mathematical Methods in the Applied Sciences, 39 (2016), 289-301.
doi: 10.1002/mma.3477. |
[33] |
X. Li and Z. Xiang,
On an attraction-repulsion chemotaxis system with a logistic source, IMA Journal of Applied Mathematics, 81 (2016), 165-198.
doi: 10.1093/imamat/hxv033. |
[34] |
J. Liu and Y. Wang,
Global existence and boundedness in a Keller-Segel-(Navier-) Stokes system with signal-dependent sensitivity, Journal of Mathematical Analysis and Applications, 447 (2017), 499-528.
doi: 10.1016/j.jmaa.2016.10.028. |
[35] |
J. Liu and Y. Wang,
Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system involving a tensor-valued sensitivity with saturation, Journal of Differential Equations, 262 (2017), 5271-5305.
doi: 10.1016/j.jde.2017.01.024. |
[36] |
P. Liu, J. Shi and Z.-A. Wang,
Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625.
doi: 10.3934/dcdsb.2013.18.2597. |
[37] |
M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner,
Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: Is there a connection?, Bulletin of Mathematical Biology, 65 (2003), 693-730.
doi: 10.1016/S0092-8240(03)00030-2. |
[38] |
A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser/Springer Basel AG, Basel, 1995. |
[39] |
N. V. Mantzaris, S. Webb and H. G. Othmer,
Mathematical modeling of tumor-induced angiogenesis, Journal of Mathematical Biology, 49 (2004), 111-187.
doi: 10.1007/s00285-003-0262-2. |
[40] |
X. Mora,
Semilinear parabolic problems define semiflows on Ck spaces, Transactions of the American Mathematical Society, 278 (1983), 21-55.
doi: 10.2307/1999300. |
[41] |
A. Quinlan and B. Straughan,
Decay bounds in a model for aggregation of microglia: Application to Alzheimer's disease senile plaques, Proceedings of the Royal Society A, 461 (2005), 2887-2897.
doi: 10.1098/rspa.2005.1483. |
[42] |
Y. Tao and M. Winkler,
Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 30 (2013), 157-178.
doi: 10.1016/j.anihpc.2012.07.002. |
[43] |
Y. Tao and M. Winkler,
Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Zeitschrift für angewandte Mathematik und Physik, 66 (2015), 2555-2573.
doi: 10.1007/s00033-015-0541-y. |
[44] |
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), Art. 138, 23 pp.
doi: 10.1007/s00033-016-0732-1. |
[45] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Communications in Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[46] |
I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein,
Bacterial swimming and oxygen transport near contact lines, Proceedings of the National Academy of Sciences of the United States of America, 102 (2005), 2277-2282.
doi: 10.1073/pnas.0406724102. |
[47] |
R. Tyson, S. R. Lubkin and J. D. Murray,
Model and analysis of chemotactic bacterial patterns in a liquid medium, Journal of Mathematical Biology, 38 (1999), 359-375.
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