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, as well as its reduced one-dimensional version. These equations are expressed in terms of the slopes u₁=hₓ u 1 = h x and u₂=hy u 2 = h y to establish the existence of global, connected attractors for both equations. Since unique solutions are guaranteed for initial conditions in Ḣ²ₚₑᵣ H ˙ p e r 2 , we consider the solution operator S(t): Ḣ²ₚₑᵣ → Ḣ²ₚₑᵣ S ( t ) : H ˙ p e r 2 → H ˙ p e r 2 , to gain our results. We prove the necessary continuity, dissipation and compactness properties.