We consider a model for the flow of a mixture of two homogeneous and incompressible fluids in a two-dimensional bounded domain. The model consists of a Navier–Stokes equation governing the fluid velocity coupled with a convective Cahn–Hilliard equation for the relative density of atoms of one of the fluids. Endowing the system with suitable boundary and initial conditions, we analyze the asymptotic behavior of its solutions. First, we prove that the initial and boundary value problem generates a strongly continuous semigroup on a suitable phase-space which possesses the global attractor $\mathcal{A}$. Then we establish the existence of an exponential attractors $\mathcal{E}$. Thus $\mathcal{A}$ has finite fractal dimension. This dimension is then estimated from above in terms of the physical parameters. Moreover, assuming the potential to be real analytic and in absence of volume forces, we demonstrate that each trajectory converges to a single equilibrium. We also obtain a convergence rate estimate in the phase-space metric.

Keywords: Navier–Stokes equations, Incompressible fluids, Cahn–Hilliard equations, Two-phase flows, Global attractors, Exponential attractors, Fractal dimension, Convergence to equilibria

@article{AIHPC_2010__27_1_401_0, author = {Gal, Ciprian G. and Grasselli, Maurizio}, title = {Asymptotic behavior of a {Cahn{\textendash}Hilliard{\textendash}Navier{\textendash}Stokes} system in {2D}}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {401--436}, publisher = {Elsevier}, volume = {27}, number = {1}, year = {2010}, doi = {10.1016/j.anihpc.2009.11.013}, mrnumber = {2580516}, zbl = {1184.35055}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2009.11.013/} }

TY - JOUR AU - Gal, Ciprian G. AU - Grasselli, Maurizio TI - Asymptotic behavior of a Cahn–Hilliard–Navier–Stokes system in 2D JO - Annales de l'I.H.P. Analyse non linéaire PY - 2010 SP - 401 EP - 436 VL - 27 IS - 1 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2009.11.013/ DO - 10.1016/j.anihpc.2009.11.013 LA - en ID - AIHPC_2010__27_1_401_0 ER -

%0 Journal Article %A Gal, Ciprian G. %A Grasselli, Maurizio %T Asymptotic behavior of a Cahn–Hilliard–Navier–Stokes system in 2D %J Annales de l'I.H.P. Analyse non linéaire %D 2010 %P 401-436 %V 27 %N 1 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2009.11.013/ %R 10.1016/j.anihpc.2009.11.013 %G en %F AIHPC_2010__27_1_401_0

Gal, Ciprian G.; Grasselli, Maurizio. Asymptotic behavior of a Cahn–Hilliard–Navier–Stokes system in 2D. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 1, pp. 401-436. doi : 10.1016/j.anihpc.2009.11.013. http://www.numdam.org/articles/10.1016/j.anihpc.2009.11.013/

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