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
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show all references

References:
[1]

Indiana Univ. Math. J., 57 (2008), 659-698. doi: 10.1512/iumj.2008.57.3391.  Google Scholar

[2]

Arch. Ration. Mech. Anal., 194 (2009), 463-506. doi: 10.1007/s00205-008-0160-2.  Google Scholar

[3]

Theoret. Comput. Fluid Dynam., 1 (1990), 303-325. Google Scholar

[4]

Physica D, 138 (2000), 360-392. doi: 10.1016/S0167-2789(99)00206-7.  Google Scholar

[5]

Pysica D (Applied Physics), 32 (1999), 1119-1123. Google Scholar

[6]

Asymptotic Anal., 20 (1999), 175-212.  Google Scholar

[7]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 225-259. doi: 10.1016/S0294-1449(00)00063-9.  Google Scholar

[8]

Computer and Fluids, 31 (2002), 41-68. Google Scholar

[9]

Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.  Google Scholar

[10]

Nonlinearity, 25 (2012), 3211-3234. doi: 10.1088/0951-7715/25/11/3211.  Google Scholar

[11]

Series Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1999. doi: 10.1137/1.9781611971088.  Google Scholar

[12]

Math. Models Methods Appl. Sci., 20 (2010), 1129-1160. 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]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436. doi: 10.1016/j.anihpc.2009.11.013.  Google Scholar

[15]

Discrete Contin. Dyn. Syst., 28 (2010), 1-39. doi: 10.3934/dcds.2010.28.1.  Google Scholar

[16]

Chin. Ann. Math. Ser. B, 31 (2010), 655-678. doi: 10.1007/s11401-010-0603-6.  Google Scholar

[17]

Math. Models Methods Appl. Sci., 6 (1996), 8-15. doi: 10.1142/S0218202596000341.  Google Scholar

[18]

SIAM J. Control Optim., 52 (2014), 747-772. doi: 10.1137/120865628.  Google Scholar

[19]

Rev. Modern Phys., 49 (1977), 435-479. Google Scholar

[20]

Russian J. Numer. Anal. math. Modelling, 16 (2001), 247-259. doi: 10.1515/rnam-2001-0305.  Google Scholar

[21]

Nonlinear Anal., 60 (2005), 1485-1508. doi: 10.1016/j.na.2004.11.010.  Google Scholar

[22]

Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[23]

Springer-Verlag, New York, 1970.  Google Scholar

[24]

submitted, 2014. Google Scholar

[25]

J. Phys. Condens. Matter, 9 (1997), 6119-6157. Google Scholar

[26]

Tellus, 33 (1981), 321-339.  Google Scholar

[27]

Q. J. R. Meteorol. Soc., 113 (1987), 1311-1328. Google Scholar

[28]

volume 68, Appl. Math. Sci., Springer-Verlag, New York, second edition, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[29]

SIAM J. Control Optim., 41 (2002), 583-606. doi: 10.1137/S0363012901385769.  Google Scholar

[30]

Nonlinear Anal., 51 (2002), 509-536. doi: 10.1016/S0362-546X(01)00843-4.  Google Scholar

[31]

Nonlinear Anal., 52 (2003), 1853-1866. doi: 10.1016/S0362-546X(02)00161-X.  Google Scholar

[32]

Nonlinear Anal., 52 (2003), 1911-1931. doi: 10.1016/S0362-546X(02)00282-1.  Google Scholar

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