Effect of Space Dimensions on Equilibrium Solutions of Cahn–Hilliard and Conservative Allen–Cahn Equations

In this study, we investigate the effect of space dimensions on the equilibrium solutions of the Cahn–Hilliard (CH) and conservative Allen–Cahn (CAC) equations in one, two, and three dimensions. The CH and CAC equations are fourth-order parabolic partial and second-order integro-partial differential equations, respectively. The former is used to model phase separation in binary mixtures, and the latter is used to model mean curvature flow with conserved mass. Both equations have been used for modeling various interface problems. To study the space-dimension effect on both the equations, we consider the equilibrium solution profiles for symmetric, radially symmetric, and spherically symmetric drop shapes. We highlight the different dynamics obtained from the CH and CAC equations. In particular, we find that there is a large difference between the solutions obtained from these equations in three-dimensional space.