# American Institute of Mathematical Sciences

October  2016, 21(8): 2663-2685. doi: 10.3934/dcdsb.2016067

## A Cahn-Hilliard-Navier-Stokes model with delays

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

Received  November 2015 Revised  July 2016 Published  September 2016

In this article, we study a coupled Cahn-Hilliard-Navier-Stokes model with delays in a two-dimensional domain. The model consists of the Navier-Stokes equations for the velocity, coupled with an Cahn-Hilliard model for the order (phase) parameter. We prove the existence and uniqueness of the weak and strong solution when the external force contains some delays. We also discuss the asymptotic behavior of the weak solutions and the stability of the stationary solutions.
Citation: 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
##### References:
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##### References:
 [1] T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flow,, Pysica D (Applied Physics), 32 (1999), 1119. doi: 10.1088/0022-3727/32/10/307. Google Scholar [2] G. Caginalp, An analysis of a phase field model of a free boundary,, Arch. Rational Mech. Anal., 92 (1986), 205. doi: 10.1007/BF00254827. Google Scholar [3] T. Caraballo, A. M. Márquez-Durán and J. Real, Pullback and forward attractors for a 3D LANS-$\alpha$ model with delay,, Discrete Contin. Dyn. Syst., 15 (2006), 559. doi: 10.3934/dcds.2006.15.559. Google Scholar [4] T. Caraballo and J. Real, Navier-Stokes equations with delays,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441. doi: 10.1098/rspa.2001.0807. Google Scholar [5] T. Caraballo and J. Real, Asymptotic behavior of two-dimensional Navier-Stokes equations with delays,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181. doi: 10.1098/rspa.2003.1166. Google Scholar [6] T. Caraballo and J. Real, Attractors for 2D Navier-Stokes models with delays,, J. Differential Equations, 205 (2004), 271. doi: 10.1016/j.jde.2004.04.012. Google Scholar [7] 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. doi: 10.1142/S0218202510004544. Google Scholar [8] 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. doi: 10.1016/j.anihpc.2009.11.013. Google Scholar [9] C. G. Gal and M. Grasselli, Longtime behavior for a model of homogeneous incompressible two-phase flows,, Discrete Contin. Dyn. Syst., 28 (2010), 1. doi: 10.3934/dcds.2010.28.1. Google Scholar [10] C. G. Gal and M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D,, Chin. Ann. Math. Ser. B, 31 (2010), 655. doi: 10.1007/s11401-010-0603-6. Google Scholar [11] P.C. Hohenberg and B. I. Halperin, Theory of dynamical critical phenomena,, Rev. Modern Phys., 49 (1977), 435. Google Scholar [12] T. Tachim Medjo, Pullback attractors for a non-autonomous homogeneous two-phase flow model,, J. Diff. Equa., 253 (2012), 1779. doi: 10.1016/j.jde.2012.06.004. Google Scholar [13] A. Onuki, Phase transition of fluids in shear flow,, J. Phys. Condens. Matter, (2009), 641. doi: 10.1017/CBO9780511534874.012. Google Scholar [14] T. Taniguchi, The exponential behavior of Navier-Stokes equations with time delay external force,, Discrete Contin. Dyn. Syst., 12 (2005), 997. doi: 10.3934/dcds.2005.12.997. Google Scholar [15] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, volume 68., Appl. Math. Sci., (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar
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