Citation: |
[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. |