Instability of two-phase flows: A lower bound on the dimension of the global attractor of the Cahn–Hilliard–Navier–Stokes system

We consider a model for the flow of a mixture of two viscous and incompressible fluids in a two or three dimensional channel-like domain. The model consists of the Navier–Stokes equations governing the fluid velocity coupled with a convective Cahn–Hilliard equation for the relative density of atoms of one of the fluids. We prove the instability of certain stationary solutions for such a system endowed with periodic boundary conditions on elongated domains ( 0 , 2 π / α 0 ) × ( 0 , 2 π ) or ( 0 , 2 π / α 0 ) × ( 0 , 2 π ) × ( 0 , 2 π / β 0 ) for a special class of periodic body forces, provided that α 0 and β 0 are small enough. As a consequence, we deduce a lower bound for the Hausdorff dimension of the global attractor.