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March  2010, 28(1): 1-39. doi: 10.3934/dcds.2010.28.1

Longtime behavior for a model of homogeneous incompressible two-phase flows

Ciprian G. Gal 1, and  Maurizio Grasselli 2,

1. 

Department of Mathematics, University of Missouri, Columbia, MO 65211, United States
2. 

Dipartimento di Matematica, Politecnico di Milano, 20133 Milano

Received  August 2009 Revised  January 2010 Published  April 2010

We consider a diffuse interface model for the evolution of an iso-thermal incompressible two-phase flow in a two-dimensional bounded domain. The model consists of the Navier-Stokes equation for the fluid velocity u coupled with a convective Allen-Cahn equation for the order (phase) parameter $\phi$, both endowed with suitable boundary conditions. We analyze the asymptotic behavior of the solutions within the theory of infinite-dimensional dissipative dynamical systems. We first 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 attractor $ \mathcal{E}$ which entails that $\mathcal{A}$ has finite fractal dimension. This dimension is then estimated in terms of some model parameters. Moreover, assuming the potential to be real analytic, we demonstrate that, in absence of external forces, each trajectory converges to a single equilibrium by means of a Łojasiewicz-Simon inequality. We also obtain a convergence rate estimate. Finally, we discuss the case where $\phi $ is forced to take values in a bounded interval, e.g., by a so-called singular potential.
Keywords: incompressible fluids, exponential attractors, Navier-Stokes equations, convergence to equilibria., Allen-Cahn equations, nematic liquid crystals, fractal dimension, global attractors.
Mathematics Subject Classification: Primary: 35B40, 35B41, 35K55, 35Q35, 37L30; Secondary: 76D05, 76T9.
Citation: Ciprian G. Gal, Maurizio Grasselli. Longtime behavior for a model of homogeneous incompressible two-phase flows. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 1-39. doi: 10.3934/dcds.2010.28.1
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