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

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

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.
Citation: T. Tachim Medjo. Robust control of a Cahn-Hilliard-Navier-Stokes model. Communications on Pure & 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,, \emph{Indiana Univ. Math. J.}, 57 (2008), 659. doi: 10.1512/iumj.2008.57.3391. Google Scholar

[2]

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

[3]

F. Abergel and R. Temam, On some control problems in fluid mechanics,, \emph{Theoret. Comput. Fluid Dynam.}, 1 (1990), 303. Google Scholar

[4]

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

[5]

T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flow,, \emph{Pysica D (Applied Physics)}, 32 (1999), 1119. Google Scholar

[6]

F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation,, \emph{Asymptotic Anal.}, 20 (1999), 175. Google Scholar

[7]

F. Boyer, Nonhomogeneous cahn-hilliard fluids,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 18 (2001), 225. doi: 10.1016/S0294-1449(00)00063-9. Google Scholar

[8]

F. Boyer, A theoretical and numerical model for the study of incompressible mixture flows,, \emph{Computer and Fluids}, 31 (2002), 41. Google Scholar

[9]

G. Caginalp, An analysis of a phase field model of a free boundary,, \emph{Arch. Rational Mech. Anal.}, 92 (1986), 205. doi: 10.1007/BF00254827. Google Scholar

[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,, \emph{Nonlinearity}, 25 (2012), 3211. doi: 10.1088/0951-7715/25/11/3211. Google Scholar

[11]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems,, Series Classics in Applied Mathematics, (1999). doi: 10.1137/1.9781611971088. Google Scholar

[12]

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

[13]

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

[14]

C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 27 (2010), 401. doi: 10.1016/j.anihpc.2009.11.013. Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

P. C. Hohenberg and B. I. Halperin, Theory of dynamical critical phenomena,, \emph{Rev. Modern Phys.}, 49 (1977), 435. Google Scholar

[20]

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

[21]

S. Li, Optimal controls of Boussinesq equations with state constraints,, \emph{Nonlinear Anal.}, 60 (2005), 1485. doi: 10.1016/j.na.2004.11.010. Google Scholar

[22]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems,, Birkh\, (1995). doi: 10.1007/978-1-4612-4260-4. Google Scholar

[23]

J. L. Lions, Optimal Control of Systems governed by Partial Differential Equations,, Springer-Verlag, (1970). Google Scholar

[24]

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

[25]

A. Onuki, Phase transition of fluids in shear flow,, \emph{J. Phys. Condens. Matter}, 9 (1997), 6119. Google Scholar

[26]

O. Talagrand, On the mathematics of data assimilation,, \emph{Tellus}, 33 (1981), 321. Google Scholar

[27]

O. Talagrand and P. Courtier, Variational assimilation of meteorological observations with the adjoint vorticity equations i: Theory,, \emph{Q. J. R. Meteorol. Soc.}, 113 (1987), 1311. Google Scholar

[28]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, volume 68, (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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

show all references

References:
[1]

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

[2]

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

[3]

F. Abergel and R. Temam, On some control problems in fluid mechanics,, \emph{Theoret. Comput. Fluid Dynam.}, 1 (1990), 303. Google Scholar

[4]

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

[5]

T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flow,, \emph{Pysica D (Applied Physics)}, 32 (1999), 1119. Google Scholar

[6]

F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation,, \emph{Asymptotic Anal.}, 20 (1999), 175. Google Scholar

[7]

F. Boyer, Nonhomogeneous cahn-hilliard fluids,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 18 (2001), 225. doi: 10.1016/S0294-1449(00)00063-9. Google Scholar

[8]

F. Boyer, A theoretical and numerical model for the study of incompressible mixture flows,, \emph{Computer and Fluids}, 31 (2002), 41. Google Scholar

[9]

G. Caginalp, An analysis of a phase field model of a free boundary,, \emph{Arch. Rational Mech. Anal.}, 92 (1986), 205. doi: 10.1007/BF00254827. Google Scholar

[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,, \emph{Nonlinearity}, 25 (2012), 3211. doi: 10.1088/0951-7715/25/11/3211. Google Scholar

[11]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems,, Series Classics in Applied Mathematics, (1999). doi: 10.1137/1.9781611971088. Google Scholar

[12]

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

[13]

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

[14]

C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 27 (2010), 401. doi: 10.1016/j.anihpc.2009.11.013. Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

P. C. Hohenberg and B. I. Halperin, Theory of dynamical critical phenomena,, \emph{Rev. Modern Phys.}, 49 (1977), 435. Google Scholar

[20]

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

[21]

S. Li, Optimal controls of Boussinesq equations with state constraints,, \emph{Nonlinear Anal.}, 60 (2005), 1485. doi: 10.1016/j.na.2004.11.010. Google Scholar

[22]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems,, Birkh\, (1995). doi: 10.1007/978-1-4612-4260-4. Google Scholar

[23]

J. L. Lions, Optimal Control of Systems governed by Partial Differential Equations,, Springer-Verlag, (1970). Google Scholar

[24]

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

[25]

A. Onuki, Phase transition of fluids in shear flow,, \emph{J. Phys. Condens. Matter}, 9 (1997), 6119. Google Scholar

[26]

O. Talagrand, On the mathematics of data assimilation,, \emph{Tellus}, 33 (1981), 321. Google Scholar

[27]

O. Talagrand and P. Courtier, Variational assimilation of meteorological observations with the adjoint vorticity equations i: Theory,, \emph{Q. J. R. Meteorol. Soc.}, 113 (1987), 1311. Google Scholar

[28]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, volume 68, (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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

[1]

T. Tachim Medjo. A Cahn-Hilliard-Navier-Stokes model with delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2663-2685. doi: 10.3934/dcdsb.2016067

[2]

Ciprian G. Gal. Robust exponential attractors for a conserved Cahn-Hilliard model with singularly perturbed boundary conditions. Communications on Pure & Applied Analysis, 2008, 7 (4) : 819-836. doi: 10.3934/cpaa.2008.7.819

[3]

Ciprian G. Gal, Alain Miranville. Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 113-147. doi: 10.3934/dcdss.2009.2.113

[4]

Laurence Cherfils, Madalina Petcu. On the viscous Cahn-Hilliard-Navier-Stokes equations with dynamic boundary conditions. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1419-1449. doi: 10.3934/cpaa.2016.15.1419

[5]

Jie Liao, Xiao-Ping Wang. Stability of an efficient Navier-Stokes solver with Navier boundary condition. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 153-171. doi: 10.3934/dcdsb.2012.17.153

[6]

Jean-Pierre Raymond. Stokes and Navier-Stokes equations with a nonhomogeneous divergence condition. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1537-1564. doi: 10.3934/dcdsb.2010.14.1537

[7]

Ciprian G. Gal, Maurizio Grasselli. Longtime behavior of nonlocal Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 145-179. doi: 10.3934/dcds.2014.34.145

[8]

Alain Miranville. Existence of solutions for Cahn-Hilliard type equations. Conference Publications, 2003, 2003 (Special) : 630-637. doi: 10.3934/proc.2003.2003.630

[9]

Desheng Li, Xuewei Ju. On dynamical behavior of viscous Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207

[10]

Laurence Cherfils, Alain Miranville, Sergey Zelik. On a generalized Cahn-Hilliard equation with biological applications. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2013-2026. doi: 10.3934/dcdsb.2014.19.2013

[11]

Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033

[12]

Sergey P. Degtyarev. On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions. Evolution Equations & Control Theory, 2015, 4 (4) : 391-429. doi: 10.3934/eect.2015.4.391

[13]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127

[14]

Christopher P. Grant. Grain sizes in the discrete Allen-Cahn and Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 127-146. doi: 10.3934/dcds.2001.7.127

[15]

Jie Shen, Xiaofeng Yang. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1669-1691. doi: 10.3934/dcds.2010.28.1669

[16]

Shixing Li, Dongming Yan. On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3077-3088. doi: 10.3934/dcdsb.2018301

[17]

Alain Miranville, Wafa Saoud, Raafat Talhouk. On the Cahn-Hilliard/Allen-Cahn equations with singular potentials. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3633-3651. doi: 10.3934/dcdsb.2018308

[18]

T. Tachim Medjo. The exponential behavior of a stochastic Cahn-Hilliard-Navier-Stokes model with multiplicative noise. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1117-1138. doi: 10.3934/cpaa.2019054

[19]

Bo You. Global attractor of the Cahn-Hilliard-Navier-Stokes system with moving contact lines. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2283-2298. doi: 10.3934/cpaa.2019103

[20]

Hamid Bellout, Jiří Neustupa, Patrick Penel. On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1353-1373. doi: 10.3934/dcds.2010.27.1353

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]