March  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 and 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.  Google Scholar

[2]

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F. W. Chaves-Silva and S. Guerrero, A uniform controllability result for the Keller-Segel system, Asymptot. Anal., 92 (2015), 313-338.  doi: 10.3233/ASY-141282.  Google Scholar

[8]

F. W. Chaves-Silva and 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.  Google Scholar

[9]

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

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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.  Google Scholar

[11]

M. Di FrancescoA. 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.  doi: 10.3934/dcds.2010.28.1437.  Google Scholar

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M. GunzburgerL. 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.  doi: 10.1137/0330011.  Google Scholar

[14]

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.  Google Scholar

[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.  Google Scholar

[16]

J. Jiang and 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.  Google Scholar

[17]

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

[18]

E. F. Keller and 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.  Google Scholar

[19]

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

[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.  doi: 10.1007/3-7643-7385-7_16.  Google Scholar

[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.  doi: 10.1137/S0363012998347110.  Google Scholar

[22]

C.-S. LinW.-M. Ni and 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.  Google Scholar

[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.  doi: 10.1016/j.anihpc.2011.04.005.  Google Scholar

[24]

E. Mallea-ZepedaE. Ortega-Torres and 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.  Google Scholar

[25]

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

[26]

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

[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.  doi: 10.1017/S0308210513000619.  Google Scholar

[28]

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

[29]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1967.  Google Scholar

[30]

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556.  doi: 10.1090/S0002-9947-1985-0808736-1.  Google Scholar

[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.  doi: 10.3934/dcds.2012.32.1901.  Google Scholar

[32]

M. J. TindallP. K. MainiS. 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.  doi: 10.1007/s11538-008-9322-5.  Google Scholar

[33]

R. TysonS. Lubkin and 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.  Google Scholar

[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.  Google Scholar

[35]

D. WoodwardR. TysonM. MyerscoughJ. D. MurrayE. Budrene and H. Berg, Spatiotemporal patterns generated by salmonella typhimurium, Biophysical Journal, 68 (1995), 2181-2189.  doi: 10.1016/S0006-3495(95)80400-5.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  doi: 10.1051/m2an/1993270202231.  Google Scholar

[2]

S. AgmonA. 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.  doi: 10.1002/cpa.3160120405.  Google Scholar

[3]

N. BellomoA. BellouquidY. Tao and 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.  Google Scholar

[4]

L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova, 31 (1961), 308-340.   Google Scholar

[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.  doi: 10.1142/S0218202505000947.  Google Scholar

[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.   Google Scholar

[7]

F. W. Chaves-Silva and S. Guerrero, A uniform controllability result for the Keller-Segel system, Asymptot. Anal., 92 (2015), 313-338.  doi: 10.3233/ASY-141282.  Google Scholar

[8]

F. W. Chaves-Silva and 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.  Google Scholar

[9]

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

[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.  Google Scholar

[11]

M. Di FrancescoA. 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.  doi: 10.3934/dcds.2010.28.1437.  Google Scholar

[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.  Google Scholar

[13]

M. GunzburgerL. 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.  doi: 10.1137/0330011.  Google Scholar

[14]

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.  Google Scholar

[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.  Google Scholar

[16]

J. Jiang and 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.  Google Scholar

[17]

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

[18]

E. F. Keller and 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.  Google Scholar

[19]

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

[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.  doi: 10.1007/3-7643-7385-7_16.  Google Scholar

[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.  doi: 10.1137/S0363012998347110.  Google Scholar

[22]

C.-S. LinW.-M. Ni and 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.  Google Scholar

[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.  doi: 10.1016/j.anihpc.2011.04.005.  Google Scholar

[24]

E. Mallea-ZepedaE. Ortega-Torres and 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.  Google Scholar

[25]

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

[26]

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

[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.  doi: 10.1017/S0308210513000619.  Google Scholar

[28]

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

[29]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1967.  Google Scholar

[30]

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556.  doi: 10.1090/S0002-9947-1985-0808736-1.  Google Scholar

[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.  doi: 10.3934/dcds.2012.32.1901.  Google Scholar

[32]

M. J. TindallP. K. MainiS. 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.  doi: 10.1007/s11538-008-9322-5.  Google Scholar

[33]

R. TysonS. Lubkin and 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.  Google Scholar

[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.  Google Scholar

[35]

D. WoodwardR. TysonM. MyerscoughJ. D. MurrayE. Budrene and H. Berg, Spatiotemporal patterns generated by salmonella typhimurium, Biophysical Journal, 68 (1995), 2181-2189.  doi: 10.1016/S0006-3495(95)80400-5.  Google Scholar

[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.  Google Scholar

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