3D convective Cahn--Hilliard equation
Al'p Eden, Varga Κ. Kalantarov · Communications on Pure & Applied Analysis · 2007
We consider the initial boundary value problem for the 3D convectiveCahn - Hilliard equation with periodic boundary conditions. Thisgives 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 thecompactness property of the solution semigroup we conclude theexistence of the global attractor. Restricting the dynamical systemon the absorbing ball in $\dot H_{per}^2(\Omega)$ and using thegeneral framework in Eden et. all. [5] the existence of anexponential attractor is guaranteed. This approach also gives anexplicit upper estimate of the dimension of the exponentialattractor, albeit of the global attractor.