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 $$u_1=h_{x}$$ u 1 = h x and $$u_2=h_y$$ u 2 = h y to establish the existence of global, connected attractors for both equations. Since unique solutions are guaranteed for initial conditions in $$\dot{H}^2_{per}$$ H ˙ p e r 2 , we consider the solution operator $$S(t): \dot{H}^2_{per} \rightarrow \dot{H}^2_{per}$$ 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.