Consider the classical $(2+1)$-dimensional Solid-On-Solid model above a hard wall on an $L\times L$ box of ${\mathbb{Z}}^2$. The model describes a crystal surface by assigning a non-negative integer height ${\eta }_x$ to each site $x$ in the box and 0 heights to its boundary. The probability of a surface configuration $\eta$ is proportional to $\exp (-\beta \mathscr{H}(\eta \left)\right)$, where $\beta$ is the inverse-temperature and $\mathscr{H}\left(\eta \right)$ sums the absolute values of height differences between neighboring sites. We give a full description of the shape of the SOS surface for low enough temperatures....

In this paper, we propose a new diffuse interface model for the study of three immiscible component incompressible viscous flows. The model is based on the Cahn-Hilliard free energy approach. The originality of our study lies in particular in the choice of the bulk free energy. We show that one must take care of this choice in order for the model to give physically relevant results. More precisely, we give conditions for the model to be well-posed and to satisfy algebraically and dynamically consistency...