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^V_M in V_M (the velocity has the H¹-regularity) and 𝒜^Y_M in Y_M (the velocity has the L²-regularity). Then we verify the regularity of the pullback attractors by proving that 𝒜^V_M=𝒜^Y_M, which implies the pullback asymptotic smoothing effect of the model in the sense that the solutions eventually become more regular than the initial data.