2018, 23(2): 557-571. doi: 10.3934/dcdsb.2017208

On a distributed control problem for a coupled chemotaxis-fluid model

1. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas and IMUS, Universidad de Sevilla, C/Tarfia, S/N, 41012 Sevilla, Spain

2. 

Universidad Industrial de Santander, Escuela de Matemáticas, Bucaramanga, A.A. 678, Colombia

* Corresponding author: M. Ángeles Rodríguez-Bellido

Received  February 2017 Revised  June 2017 Published  December 2017

Fund Project: The first author has been partially supported by MINECO grants MTM2012-32325 and MTM2015-69875-P (Ministerio de Economía y Competitividad, Spain) with the participation of FEDER. The second and third authors have been supported by Vicerrectoría de Investigación y Extensión of Universidad Industrial de Santander, and Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco José de Caldas, contrato Colciencias FP 44842- 157-2016.

In this paper we analyze an optimal distributed control problem where the state equations are given by a stationary chemotaxis model coupled with the Navier-Stokes equations. We consider that the movement and the interaction of cells are occurring in a smooth bounded domain of $\mathbb{R}^n,n = 2,3,$ subject to homogeneous boundary conditions. We control the system through a distributed force and a coefficient of chemotactic sensitivity, leading the chemical concentration, the cell density, and the velocity field towards a given target concentration, density and velocity, respectively. In addition to the existence of optimal solution, we derive some optimality conditions.

Citation: M. Ángeles Rodríguez-Bellido, Diego A. Rueda-Gómez, Élder J. Villamizar-Roa. On a distributed control problem for a coupled chemotaxis-fluid model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 557-571. doi: 10.3934/dcdsb.2017208
References:
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F. Abergel, E. Casas, Some optimal control problems of multistate equations appearing in fluid mechanics, RAIRO Modél. Math. Anal. Numér., 27 (1993), 223-247. doi: 10.1051/m2an/1993270202231.

[2]

S. Agmon, A. Douglis, 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. doi: 10.1002/cpa.3160120405.

[3]

N. Bellomo, A. Bellouquid, Y. Tao, M. Winkler, Toward a mathematical theory of KellerSegel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X.

[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, 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. doi: 10.1142/S0218202505000947.

[6]

M. A. Chaplain, 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, S. Guerrero, A uniform controllability result for the Keller-Segel system, Asymptot. Anal., 92 (2015), 313-338. doi: 10.3233/ASY-141282.

[8]

F. W. Chaves-Silva, S. Guerrero, A controllability result for a chemotaxis-fluid model, J. Differential Equations, 262 (2017), 4863-4905. doi: 10.1016/j.jde.2017.01.004.

[9]

M. del Pino, J. Wei, Collapsing steady states of the Keller-Segel system, Nonlinearity, 19 (2006), 661-684. doi: 10.1088/0951-7715/19/3/007.

[10]

E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics, Birkäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8843-0.

[11]

M. Di Francesco, A. Lorz, 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. doi: 10.3934/dcds.2010.28.1437.

[12]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 2001. doi: 10.1007/978-3-642-61798-0.

[13]

M. Gunzburger, L. Hou, T. Svobodny, Boundary velocity control of incompressible flow with an application to viscous drag reduction, SIAM J. Control Optim., 30 (1992), 167-181. doi: 10.1137/0330011.

[14]

T. Hillen, 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.

[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). doi: 10.1023/A:1010297424324.

[16]

J. Jiang, Y. Y. Zhang, On convergence to equilibria for a chemotaxis model with volumefilling effect, Asymptot. Anal., 65 (2009), 79-102. doi: 10.3233/ASY-2009-0948.

[17]

Y. Kabeya, W. Ni, Stationary Keller-Segel model with the linear sensitivity, Surikaisekikenkyusho Kokyuroku, (1998), 44-65.

[18]

E. F. Keller, L. A. Segel, Initiation of slide mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.

[19]

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

[20]

P. Laurençot, 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. doi: 10.1007/3-7643-7385-7_16.

[21]

H.-C. Lee, 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. doi: 10.1137/S0363012998347110.

[22]

C.-S. Lin, W.-M. Ni, I. Takagi, Large amplitude stationary solutions to a Chemotaxis system, J. Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7.

[23]

J.-G. Liu, A. Lorz, A coupled chemotaxis-fluid model: global existence, Ann. Inst. H. Poincaré. Anal. Non Linéare, 28 (2011), 643-652. doi: 10.1016/j.anihpc.2011.04.005.

[24]

E. Mallea-Zepeda, E. Ortega-Torres, E. J. Villamizar-Roa, A boundary control problem for micropolar fluids, J. Optim. Theory Appl., 169 (2016), 349-369. doi: 10.1007/s10957-016-0925-y.

[25]

N. V. Mantzaris, S. Webb, H. G. Othmer, Mathematical modeling of tumor-induced angiogenesis, J. Math. Biol., 49 (2004), 111-187. doi: 10.1007/s00285-003-0262-2.

[26]

M. Musso, J. Wei, Stationary solutions to Keller-Segel chemotaxis system, Asymptot. Anal., 49 (2006), 217-247.

[27]

A. Pistoia, G. Vaira, Steady states with unbounded mass of the Keller-Segel system, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 203-222. doi: 10.1017/S0308210513000619.

[28]

A. Potapov, T. Hillen, Metastability in chemotaxis models, J. Dynam. Differential Equations, 17 (2005), 293-330. doi: 10.1007/s10884-005-2938-3.

[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. doi: 10.1090/S0002-9947-1985-0808736-1.

[31]

Y. Tao, 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. doi: 10.3934/dcds.2012.32.1901.

[32]

M. J. Tindall, P. K. Maini, S. L. Porter, J. P. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis Ⅱ: Bacterial populations, Bulletin of Mathematical Biology, 70 (2008), 1570-1607. doi: 10.1007/s11538-008-9322-5.

[33]

R. Tyson, S. Lubkin, J. D. Murray, Model and analysis of chemotactic bacterial patterns in a liquid medium, J. Math. Biol., 38 (1999), 359-375. doi: 10.1007/s002850050153.

[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. doi: 10.1080/03605302.2011.591865.

[35]

D. Woodward, R. Tyson, M. Myerscough, J. D. Murray, E. Budrene, H. Berg, Spatiotemporal patterns generated by salmonella typhimurium, Biophysical Journal, 68 (1995), 2181-2189. doi: 10.1016/S0006-3495(95)80400-5.

[36]

X. Ye, Existence and decay of global smooth solutions to the coupled chemotaxis-fluid model, J. Math. Anal. Appl., 427 (2015), 60-73. doi: 10.1016/j.jmaa.2015.02.023.

show all references

References:
[1]

F. Abergel, E. Casas, Some optimal control problems of multistate equations appearing in fluid mechanics, RAIRO Modél. Math. Anal. Numér., 27 (1993), 223-247. doi: 10.1051/m2an/1993270202231.

[2]

S. Agmon, A. Douglis, 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. doi: 10.1002/cpa.3160120405.

[3]

N. Bellomo, A. Bellouquid, Y. Tao, M. Winkler, Toward a mathematical theory of KellerSegel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X.

[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, 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. doi: 10.1142/S0218202505000947.

[6]

M. A. Chaplain, 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, S. Guerrero, A uniform controllability result for the Keller-Segel system, Asymptot. Anal., 92 (2015), 313-338. doi: 10.3233/ASY-141282.

[8]

F. W. Chaves-Silva, S. Guerrero, A controllability result for a chemotaxis-fluid model, J. Differential Equations, 262 (2017), 4863-4905. doi: 10.1016/j.jde.2017.01.004.

[9]

M. del Pino, J. Wei, Collapsing steady states of the Keller-Segel system, Nonlinearity, 19 (2006), 661-684. doi: 10.1088/0951-7715/19/3/007.

[10]

E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics, Birkäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8843-0.

[11]

M. Di Francesco, A. Lorz, 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. doi: 10.3934/dcds.2010.28.1437.

[12]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 2001. doi: 10.1007/978-3-642-61798-0.

[13]

M. Gunzburger, L. Hou, T. Svobodny, Boundary velocity control of incompressible flow with an application to viscous drag reduction, SIAM J. Control Optim., 30 (1992), 167-181. doi: 10.1137/0330011.

[14]

T. Hillen, 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.

[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). doi: 10.1023/A:1010297424324.

[16]

J. Jiang, Y. Y. Zhang, On convergence to equilibria for a chemotaxis model with volumefilling effect, Asymptot. Anal., 65 (2009), 79-102. doi: 10.3233/ASY-2009-0948.

[17]

Y. Kabeya, W. Ni, Stationary Keller-Segel model with the linear sensitivity, Surikaisekikenkyusho Kokyuroku, (1998), 44-65.

[18]

E. F. Keller, L. A. Segel, Initiation of slide mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.

[19]

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

[20]

P. Laurençot, 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. doi: 10.1007/3-7643-7385-7_16.

[21]

H.-C. Lee, 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. doi: 10.1137/S0363012998347110.

[22]

C.-S. Lin, W.-M. Ni, I. Takagi, Large amplitude stationary solutions to a Chemotaxis system, J. Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7.

[23]

J.-G. Liu, A. Lorz, A coupled chemotaxis-fluid model: global existence, Ann. Inst. H. Poincaré. Anal. Non Linéare, 28 (2011), 643-652. doi: 10.1016/j.anihpc.2011.04.005.

[24]

E. Mallea-Zepeda, E. Ortega-Torres, E. J. Villamizar-Roa, A boundary control problem for micropolar fluids, J. Optim. Theory Appl., 169 (2016), 349-369. doi: 10.1007/s10957-016-0925-y.

[25]

N. V. Mantzaris, S. Webb, H. G. Othmer, Mathematical modeling of tumor-induced angiogenesis, J. Math. Biol., 49 (2004), 111-187. doi: 10.1007/s00285-003-0262-2.

[26]

M. Musso, J. Wei, Stationary solutions to Keller-Segel chemotaxis system, Asymptot. Anal., 49 (2006), 217-247.

[27]

A. Pistoia, G. Vaira, Steady states with unbounded mass of the Keller-Segel system, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 203-222. doi: 10.1017/S0308210513000619.

[28]

A. Potapov, T. Hillen, Metastability in chemotaxis models, J. Dynam. Differential Equations, 17 (2005), 293-330. doi: 10.1007/s10884-005-2938-3.

[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. doi: 10.1090/S0002-9947-1985-0808736-1.

[31]

Y. Tao, 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. doi: 10.3934/dcds.2012.32.1901.

[32]

M. J. Tindall, P. K. Maini, S. L. Porter, J. P. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis Ⅱ: Bacterial populations, Bulletin of Mathematical Biology, 70 (2008), 1570-1607. doi: 10.1007/s11538-008-9322-5.

[33]

R. Tyson, S. Lubkin, J. D. Murray, Model and analysis of chemotactic bacterial patterns in a liquid medium, J. Math. Biol., 38 (1999), 359-375. doi: 10.1007/s002850050153.

[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. doi: 10.1080/03605302.2011.591865.

[35]

D. Woodward, R. Tyson, M. Myerscough, J. D. Murray, E. Budrene, H. Berg, Spatiotemporal patterns generated by salmonella typhimurium, Biophysical Journal, 68 (1995), 2181-2189. doi: 10.1016/S0006-3495(95)80400-5.

[36]

X. Ye, Existence and decay of global smooth solutions to the coupled chemotaxis-fluid model, J. Math. Anal. Appl., 427 (2015), 60-73. doi: 10.1016/j.jmaa.2015.02.023.

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