# American Institute of Mathematical Sciences

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. [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. [3] F. Abergel and R. Temam, On some control problems in fluid mechanics,, \emph{Theoret. Comput. Fluid Dynam.}, 1 (1990), 303. [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. [5] T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flow,, \emph{Pysica D (Applied Physics)}, 32 (1999), 1119. [6] F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation,, \emph{Asymptotic Anal.}, 20 (1999), 175. [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. [8] F. Boyer, A theoretical and numerical model for the study of incompressible mixture flows,, \emph{Computer and Fluids}, 31 (2002), 41. [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. [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. [11] I. Ekeland and R. Temam, Convex Analysis and Variational Problems,, Series Classics in Applied Mathematics, (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,, \emph{Math. Models Methods Appl. Sci.}, 20 (2010), 1129. 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,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 27 (2010), 401. 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,, \emph{Discrete Contin. Dyn. Syst.}, 28 (2010), 1. doi: 10.3934/dcds.2010.28.1. [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. [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. [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. [19] P. C. Hohenberg and B. I. Halperin, Theory of dynamical critical phenomena,, \emph{Rev. Modern Phys.}, 49 (1977), 435. [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. [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. [22] X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems,, Birkh\, (1995). doi: 10.1007/978-1-4612-4260-4. [23] J. L. Lions, Optimal Control of Systems governed by Partial Differential Equations,, Springer-Verlag, (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,, \emph{J. Phys. Condens. Matter}, 9 (1997), 6119. [26] O. Talagrand, On the mathematics of data assimilation,, \emph{Tellus}, 33 (1981), 321. [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. [28] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, volume 68, (1997). doi: 10.1007/978-1-4612-0645-3. [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. [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. [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. [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.

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. [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. [3] F. Abergel and R. Temam, On some control problems in fluid mechanics,, \emph{Theoret. Comput. Fluid Dynam.}, 1 (1990), 303. [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. [5] T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flow,, \emph{Pysica D (Applied Physics)}, 32 (1999), 1119. [6] F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation,, \emph{Asymptotic Anal.}, 20 (1999), 175. [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. [8] F. Boyer, A theoretical and numerical model for the study of incompressible mixture flows,, \emph{Computer and Fluids}, 31 (2002), 41. [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. [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. [11] I. Ekeland and R. Temam, Convex Analysis and Variational Problems,, Series Classics in Applied Mathematics, (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,, \emph{Math. Models Methods Appl. Sci.}, 20 (2010), 1129. 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,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 27 (2010), 401. 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,, \emph{Discrete Contin. Dyn. Syst.}, 28 (2010), 1. doi: 10.3934/dcds.2010.28.1. [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. [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. [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. [19] P. C. Hohenberg and B. I. Halperin, Theory of dynamical critical phenomena,, \emph{Rev. Modern Phys.}, 49 (1977), 435. [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. [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. [22] X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems,, Birkh\, (1995). doi: 10.1007/978-1-4612-4260-4. [23] J. L. Lions, Optimal Control of Systems governed by Partial Differential Equations,, Springer-Verlag, (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,, \emph{J. Phys. Condens. Matter}, 9 (1997), 6119. [26] O. Talagrand, On the mathematics of data assimilation,, \emph{Tellus}, 33 (1981), 321. [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. [28] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, volume 68, (1997). doi: 10.1007/978-1-4612-0645-3. [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. [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. [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. [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.
 [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] Alain Miranville, Wafa Saoud, Raafat Talhouk. On the Cahn-Hilliard/Allen-Cahn equations with singular potentials. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-19. doi: 10.3934/dcdsb.2018308 [17] 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 [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