Pullback attractors for a non-autonomous Cahn-Hilliard-Navier-Stokes system in 2D

This article studies the pullback asymptotic behavior of solutions for a non-autonomous Cahn-Hilliard-Navier-Stokes (CH-NS) system in a two-dimensional domain. We prove the existence of pullback attractors A(VM) in V-M (the velocity has the H-1-regularity) and A(YM) in Y-M (the velocity has the L-2-regularity). Then we verify the regularity of the pullback attractors by proving that A(YM) = A(YM), which implies the pullback asymptotic smoothing effect of the model in the sense that the solutions eventually become more regular than the initial data.