3D convective Cahn--Hilliard equation

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.