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November  2016, 15(6): 2075-2101. doi: 10.3934/cpaa.2016028

Robust control of a Cahn-Hilliard-Navier-Stokes model

T. Tachim Medjo 1,

1. 

Department of Mathematics, Florida International University, DM413B, University Park, Miami, Florida 33199, United States

Received  November 2015 Revised  March 2016 Published  September 2016

We study in this article a class of robust control problems associated with a coupled Cahn-Hilliard-Navier-Stokes model in a two dimensional bounded domain. The model consists of the Navier-Stokes equations for the velocity, coupled with the Cahn-Hilliard model for the order (phase) parameter. We prove the existence and uniqueness of solutions and we derive a first-order necessary optimality condition for these robust control problems.
Keywords: optimality condition., Cahn-Hilliard, Navier-Stokes, Robust control.
Mathematics Subject Classification: 93C05, 93B50, 93C3.
Citation: T. Tachim Medjo. Robust control of a Cahn-Hilliard-Navier-Stokes model. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2075-2101. doi: 10.3934/cpaa.2016028
References:
[1]

H. Abels, On a diffuse interface model for a two-phase flow of compressible viscous fluids, Indiana Univ. Math. J., 57 (2008), 659-698. doi: 10.1512/iumj.2008.57.3391.

[2]

H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Ration. Mech. Anal., 194 (2009), 463-506. doi: 10.1007/s00205-008-0160-2.

[3]

F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoret. Comput. Fluid Dynam., 1 (1990), 303-325.

[4]

T. Bewley, R. Temam and M. Ziane, A general framework for robust control in fluid mechanics, Physica D, 138 (2000), 360-392. doi: 10.1016/S0167-2789(99)00206-7.

[5]

T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flow, Pysica D (Applied Physics), 32 (1999), 1119-1123.

[6]

F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptotic Anal., 20 (1999), 175-212.

[7]

F. Boyer, Nonhomogeneous cahn-hilliard fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 225-259. doi: 10.1016/S0294-1449(00)00063-9.

[8]

F. Boyer, A theoretical and numerical model for the study of incompressible mixture flows, Computer and Fluids, 31 (2002), 41-68.

[9]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.

[10]

C. Cao and C. G. Gal, Global solutions for the 2D NS-CH model for a two-phase flow of viscous, incompressible fluids with mixed partial viscosity and mobility, Nonlinearity, 25 (2012), 3211-3234. doi: 10.1088/0951-7715/25/11/3211.

[11]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Series Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1999. doi: 10.1137/1.9781611971088.

[12]

E. Feireisl, H. Petzeltová, E. Rocca and G. Schimperna, Analysis of a phase-field model for two-phase compressible fluids, Math. Models Methods Appl. Sci., 20 (2010), 1129-1160. doi: 10.1142/S0218202510004544.

[13]

S. Frigeri, E. Rocca and J. Sprekels, Optimal distributed control of a nonlocal Cahn-Hilliard/Navier-stokes system in 2D,, arXiv:1411.1627., (). 

[14]

C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436. doi: 10.1016/j.anihpc.2009.11.013.

[15]

C. G. Gal and M. Grasselli, Longtime behavior for a model of homogeneous incompressible two-phase flows, Discrete Contin. Dyn. Syst., 28 (2010), 1-39. doi: 10.3934/dcds.2010.28.1.

[16]

C. G. Gal and M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D, Chin. Ann. Math. Ser. B, 31 (2010), 655-678. doi: 10.1007/s11401-010-0603-6.

[17]

M. E. Gurtin, D. Polignone and J. Vinals, Two-phase binary fluid and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6 (1996), 8-15. doi: 10.1142/S0218202596000341.

[18]

M. Hintermüller and D. Wegner, Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system, SIAM J. Control Optim., 52 (2014), 747-772. doi: 10.1137/120865628.

[19]

P. C. Hohenberg and B. I. Halperin, Theory of dynamical critical phenomena, Rev. Modern Phys., 49 (1977), 435-479.

[20]

F. X. LeDimet and V. Shutyaev, On data assimilation for quasilinear parabolic problems, Russian J. Numer. Anal. math. Modelling, 16 (2001), 247-259. doi: 10.1515/rnam-2001-0305.

[21]

S. Li, Optimal controls of Boussinesq equations with state constraints, Nonlinear Anal., 60 (2005), 1485-1508. doi: 10.1016/j.na.2004.11.010.

[22]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.

[23]

J. L. Lions, Optimal Control of Systems governed by Partial Differential Equations, Springer-Verlag, New York, 1970.

[24]

T. Tachim Medjo, Optimal control of a Cahn-Hilliard-Navier-Stokes model with state constraints, submitted, 2014.

[25]

A. Onuki, Phase transition of fluids in shear flow, J. Phys. Condens. Matter, 9 (1997), 6119-6157.

[26]

O. Talagrand, On the mathematics of data assimilation, Tellus, 33 (1981), 321-339.

[27]

O. Talagrand and P. Courtier, Variational assimilation of meteorological observations with the adjoint vorticity equations i: Theory, Q. J. R. Meteorol. Soc., 113 (1987), 1311-1328.

[28]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, volume 68, Appl. Math. Sci., Springer-Verlag, New York, second edition, 1997. doi: 10.1007/978-1-4612-0645-3.

[29]

G. Wang, Optimal controls of 3-dimensional Navier-Stokes equations with state constraints, SIAM J. Control Optim., 41 (2002), 583-606. doi: 10.1137/S0363012901385769.

[30]

G. Wang, Pontryagin maximum principle of optimal control governed by fluid dynamic systems with two point boundary state constraint, Nonlinear Anal., 51 (2002), 509-536. doi: 10.1016/S0362-546X(01)00843-4.

[31]

G. Wang, Pontryagin's maximum principle for optimal control of the stationary Navier-Stokes equations, Nonlinear Anal., 52 (2003), 1853-1866. doi: 10.1016/S0362-546X(02)00161-X.

[32]

G. Wang and L. Wang, Maximum principle of state-constrained optimal control governed by fluid dynamic systems, Nonlinear Anal., 52 (2003), 1911-1931. doi: 10.1016/S0362-546X(02)00282-1.

show all references

References:
[1]

H. Abels, On a diffuse interface model for a two-phase flow of compressible viscous fluids, Indiana Univ. Math. J., 57 (2008), 659-698. doi: 10.1512/iumj.2008.57.3391.

[2]

H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Ration. Mech. Anal., 194 (2009), 463-506. doi: 10.1007/s00205-008-0160-2.

[3]

F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoret. Comput. Fluid Dynam., 1 (1990), 303-325.

[4]

T. Bewley, R. Temam and M. Ziane, A general framework for robust control in fluid mechanics, Physica D, 138 (2000), 360-392. doi: 10.1016/S0167-2789(99)00206-7.

[5]

T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flow, Pysica D (Applied Physics), 32 (1999), 1119-1123.

[6]

F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptotic Anal., 20 (1999), 175-212.

[7]

F. Boyer, Nonhomogeneous cahn-hilliard fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 225-259. doi: 10.1016/S0294-1449(00)00063-9.

[8]

F. Boyer, A theoretical and numerical model for the study of incompressible mixture flows, Computer and Fluids, 31 (2002), 41-68.

[9]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.

[10]

C. Cao and C. G. Gal, Global solutions for the 2D NS-CH model for a two-phase flow of viscous, incompressible fluids with mixed partial viscosity and mobility, Nonlinearity, 25 (2012), 3211-3234. doi: 10.1088/0951-7715/25/11/3211.

[11]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Series Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1999. doi: 10.1137/1.9781611971088.

[12]

E. Feireisl, H. Petzeltová, E. Rocca and G. Schimperna, Analysis of a phase-field model for two-phase compressible fluids, Math. Models Methods Appl. Sci., 20 (2010), 1129-1160. doi: 10.1142/S0218202510004544.

[13]

S. Frigeri, E. Rocca and J. Sprekels, Optimal distributed control of a nonlocal Cahn-Hilliard/Navier-stokes system in 2D,, arXiv:1411.1627., (). 

[14]

C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436. doi: 10.1016/j.anihpc.2009.11.013.

[15]

C. G. Gal and M. Grasselli, Longtime behavior for a model of homogeneous incompressible two-phase flows, Discrete Contin. Dyn. Syst., 28 (2010), 1-39. doi: 10.3934/dcds.2010.28.1.

[16]

C. G. Gal and M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D, Chin. Ann. Math. Ser. B, 31 (2010), 655-678. doi: 10.1007/s11401-010-0603-6.

[17]

M. E. Gurtin, D. Polignone and J. Vinals, Two-phase binary fluid and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6 (1996), 8-15. doi: 10.1142/S0218202596000341.

[18]

M. Hintermüller and D. Wegner, Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system, SIAM J. Control Optim., 52 (2014), 747-772. doi: 10.1137/120865628.

[19]

P. C. Hohenberg and B. I. Halperin, Theory of dynamical critical phenomena, Rev. Modern Phys., 49 (1977), 435-479.

[20]

F. X. LeDimet and V. Shutyaev, On data assimilation for quasilinear parabolic problems, Russian J. Numer. Anal. math. Modelling, 16 (2001), 247-259. doi: 10.1515/rnam-2001-0305.

[21]

S. Li, Optimal controls of Boussinesq equations with state constraints, Nonlinear Anal., 60 (2005), 1485-1508. doi: 10.1016/j.na.2004.11.010.

[22]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.

[23]

J. L. Lions, Optimal Control of Systems governed by Partial Differential Equations, Springer-Verlag, New York, 1970.

[24]

T. Tachim Medjo, Optimal control of a Cahn-Hilliard-Navier-Stokes model with state constraints, submitted, 2014.

[25]

A. Onuki, Phase transition of fluids in shear flow, J. Phys. Condens. Matter, 9 (1997), 6119-6157.

[26]

O. Talagrand, On the mathematics of data assimilation, Tellus, 33 (1981), 321-339.

[27]

O. Talagrand and P. Courtier, Variational assimilation of meteorological observations with the adjoint vorticity equations i: Theory, Q. J. R. Meteorol. Soc., 113 (1987), 1311-1328.

[28]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, volume 68, Appl. Math. Sci., Springer-Verlag, New York, second edition, 1997. doi: 10.1007/978-1-4612-0645-3.

[29]

G. Wang, Optimal controls of 3-dimensional Navier-Stokes equations with state constraints, SIAM J. Control Optim., 41 (2002), 583-606. doi: 10.1137/S0363012901385769.

[30]

G. Wang, Pontryagin maximum principle of optimal control governed by fluid dynamic systems with two point boundary state constraint, Nonlinear Anal., 51 (2002), 509-536. doi: 10.1016/S0362-546X(01)00843-4.

[31]

G. Wang, Pontryagin's maximum principle for optimal control of the stationary Navier-Stokes equations, Nonlinear Anal., 52 (2003), 1853-1866. doi: 10.1016/S0362-546X(02)00161-X.

[32]

G. Wang and L. Wang, Maximum principle of state-constrained optimal control governed by fluid dynamic systems, Nonlinear Anal., 52 (2003), 1911-1931. doi: 10.1016/S0362-546X(02)00282-1.

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