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December  2007, 6(4): 1075-1086. doi: 10.3934/cpaa.2007.6.1075

3D convective Cahn--Hilliard equation

1. 

Department of Mathematics, Bogaziçi University, TUBITAK Feza Gürsey Institute, for Basic Sciences, Istanbul, Turkey

2. 

Department of Mathematics, Koç University, Istanbul, Turkey

Received  December 2006 Revised  May 2007 Published  September 2007

We consider the initial boundary value problem for the 3D convective Cahn - Hilliard equation with periodic boundary conditions. This gives rise to a continuous dynamical system on $\dot L^2(\Omega)$. Absorbing balls in $\dot L^2(\Omega), \dot H_{per}^1(\Omega)$ and $\dot H_{per}^2(\Omega)$ are shown to exist. Combining with the compactness property of the solution semigroup we conclude the existence of the global attractor. Restricting the dynamical system on the absorbing ball in $\dot H_{per}^2(\Omega)$ and using the general framework in Eden et. all. [5] the existence of an exponential attractor is guaranteed. This approach also gives an explicit upper estimate of the dimension of the exponential attractor, albeit of the global attractor.
Citation: Alp Eden, Varga K. Kalantarov. 3D convective Cahn--Hilliard equation. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1075-1086. doi: 10.3934/cpaa.2007.6.1075
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