Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

On a distributed control problem for a coupled chemotaxis-fluid model
Page number are going to be assigned later 2017

doi:10.3934/dcdsb.2017208      Abstract        References        Full text (435.7K)      

M. Ángeles Rodríguez-Bellido - Departamento de Ecuaciones Diferenciales y Análisis Numérico , Facultad de Matemáticas and IMUS, Universidad de Sevilla, C/ Tarfia, S/N, 41012 Sevilla, Spain (email)
Diego A. Rueda-Gómez - Universidad Industrial de Santander, Escuela de Matemáticas, Bucaramanga, A.A. 678, Colombia (email)
Élder J. Villamizar-Roa - Universidad Industrial de Santander, Escuela de Matemáticas, Bucaramanga, A.A. 678, Colombia (email)

1 F. Abergel and E. Casas, Some optimal control problems of multistate equations appearing in fluid mechanics, RAIRO Modél. Math. Anal. Numér., 27 (1993), 223-247.       
2 S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I., Commun. Pure Appl. Math., 12 (1959), 623-727.       
3 N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.       
4 L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova, 31 (1961), 308-340.       
5 M. A. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Math. Models Methods Appl. Sci., 15 (2005), 1685-1734.       
6 M. A. Chaplain and A. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor, IMA J. Math. Appl. Med. Biol., 10 (1993), 149-168.
7 F. W. Chaves-Silva and S. Guerrero, A uniform controllability result for the Keller-Segel system, Asymptot. Anal., 92 (2015), 313-338.       
8 F. W. Chaves-Silva and S. Guerrero, A controllability result for a chemotaxis-fluid model, J. Differential Equations, 262 (2017), 4863-4905.       
9 M. del Pino and J. Wei, Collapsing steady states of the Keller-Segel system, Nonlinearity, 19 (2006), 661-684.       
10 E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics, Birkäuser Verlag, Basel, 2009.       
11 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., 28 (2010), 1437-1453.       
12 D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 2001.       
13 M. Gunzburger, L. Hou and T. Svobodny, Boundary velocity control of incompressible flow with an application to viscous drag reduction, SIAM J. Control Optim., 30 (1992), 167-181.       
14 T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.       
15 A. A. Illarionov, Optimal boundary control of the steady flow of a viscous nonhomogeneous incompressible fluid, Math. Notes, 69 (2001), 614-624(Translated from Mat. Zametki, 69 (2001), 666-678).       
16 J. Jiang and Y. Y. Zhang, On convergence to equilibria for a chemotaxis model with volume-filling effect, Asymptot. Anal., 65 (2009), 79-102.       
17 Y. Kabeya and W. Ni, Stationary Keller-Segel model with the linear sensitivity, Sūrikaisekikenkyūsho Kōkyūroku, (1998), 44-65.       
18 E. F. Keller and L. A. Segel, Initiation of slide mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
19 E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.
20 P. Laurençot and D. Wrzosek, A chemotaxis model with threshold density and degenerate diffusion, Nonlinear Elliptic and Parabolic Problems, Progr. Nonlinear Differential Equations Appl., 64 (2005), 273-290.       
21 H.-C. Lee and O. Y. Imanuvilov, Analysis of Newmann boundary optimal control problems for the stationary Boussinesq equations including solid media, SIAM J. Control Optim., 39 (2000), 457-477.       
22 C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a Chemotaxis system, J. Differential Equations, 72 (1988), 1-27.       
23 J.-G. Liu and A. Lorz, A coupled chemotaxis-fluid model: global existence, Ann. Inst. H. Poincaré. Anal. Non Linéare, 28 (2011), 643-652.       
24 E. Mallea-Zepeda, E. Ortega-Torres and E. J. Villamizar-Roa, A boundary control problem for micropolar fluids, J. Optim. Theory Appl., 169 (2016), 349-369.       
25 N. V. Mantzaris, S. Webb and H. G. Othmer, Mathematical modeling of tumor-induced angiogenesis, J. Math. Biol., 49 (2004), 111-187.       
26 M. Musso and J. Wei, Stationary solutions to Keller-Segel chemotaxis system, Asymptot. Anal., 49 (2006), 217-247.       
27 A. Pistoia and G. Vaira, Steady states with unbounded mass of the Keller-Segel system, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 203-222.       
28 A. Potapov and T. Hillen, Metastability in chemotaxis models, J. Dynam. Differential Equations, 17 (2005), 293-330.       
29 M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967.       
30 R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556.       
31 Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst., 32 (2012), 1901-1914.       
32 M. J. Tindall, P. K. Maini, S. L. Porter and J. P. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis II: Bacterial populations, Bulletin of Mathematical Biology, 70 (2008), 1570-1607.       
33 R. Tyson, S. Lubkin and J. D. Murray, Model and analysis of chemotactic bacterial patterns in a liquid medium, J. Math. Biol., 38 (1999), 359-375.       
34 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.       
35 D. Woodward, R. Tyson, M. Myerscough, J. D. Murray, E. Budrene and H. Berg, Spatio-temporal patterns generated by salmonella typhimurium, Biophysical Journal, 68 (1995), 2181-2189.
36 X. Ye, Existence and decay of global smooth solutions to the coupled chemotaxis-fluid model, J. Math. Anal. Appl., 427 (2015), 60-73.       

Go to top