Global attractors of sixth order PDEs describing the faceting of growing surfaces

A spatially two-dimensional sixth order PDE describing the evolution of a growing crystalline surface h(x, y, t) that undergoes faceting is considered with periodic boundary conditions, such as its reduced onedimensional version. These equation are expressed in terms of the slopes u1 = hx and u2 = hy to establish the existence of global, connected attractors for both of the equations. Since unique solutions are guaranteed for initial conditions in Ḣ per, we consider the solution operator S(t) : Ḣ per → Ḣ per, to gain the results. We prove the necessary continuity, dissipation and compactness properties.