Long-time dynamics of a bulk-surface convective Cahn--Hilliard system: Pullback attractors and convergence to equilibrium

We study the long-time dynamics of a bulk-surface convective Cahn--Hilliard system describing phase separation processes with bulk-surface interaction. The presence of convection terms leads to a non-autonomous dynamical system and prevents the associated free energy from being a Lyapunov functional, which makes the analysis of the asymptotic behavior considerably more challenging. First, we establish an instantaneous regularization property for weak solutions. Next, interpreting the evolution as a continuous two-parameter process, we prove the existence of a minimal pullback attractor. Finally, under suitable decay assumptions on the velocity fields, we show that every solution converges as $t\to\infty$ to a single steady state. The proof of this convergence relies on the {\L}ojasiewicz--Simon inequality combined with customized decay estimates that compensate for the lack of a monotone energy functional.