# American Institute of Mathematical Sciences

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