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A convective Cahn-Hilliard type equation of sixth order that describes the faceting of a growing surface is considered with periodic boundary conditions. By using a Galerkin approach the existence of weak solutions to this sixth order partial differential equation is established in L 2 (0, T ;Ḣ 3 per). Furthermore stronger regularity results have been derived and these are used to prove uniqueness of the solutions. Additionally a numerical study shows that solutions behave similarly as for the better known convective Cahn-Hilliard equation. The transition from coarsening to roughening is analyzed, indicating that the characteristic length scale decreases logarithmically with increasing deposition rate.

New types of stationary solutions of a one-dimensional driven sixthorder Cahn-Hilliard type equation that arises as a model for epitaxially growing nano-structures such as quantum dots, are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms. This method yields analytical expressions for far-field behavior as well as the widths of the humps of these spatially non-monotone solutions in the limit of small driving force strength which is the deposition rate in case of epitaxial growth. These solutions extend the family of the monotone kink and antikink solutions. The hump spacing is related to solutions of the Lambert W function.Using phase space analysis for the corresponding fifth-order dynamical system, we use a numerical technique that enables the efficient and accurate tracking of the solution branches, where the asymptotic solutions are used as initial input.Additionally, our approach is first demonstrated for the related but simpler driven fourth-order Cahn-Hilliard equation, also known as the convective Cahn-Hilliard equation.

In this paper we study a sixth order Cahn-Hilliard type equation that arises as a model for the faceting of a growing surface. We show global in time existence of weak solutions and uniform in time a priori estimates in the H 3 norm. These bounds enable us to show the uniqueness of weak solutions.

In this paper we revisit models for the description of the evolution of crystalline films with anisotropic surface energies. We prove equivalences of symmetry properties of anisotropic surface energy models commonly used in the literature. Then we systematically develop a framework for the derivation of surface diffusion models for the self-assembly of quantum dots during Stranski-Krastanov growth that include surface energies also with large anisotropy as well as the effect of wetting energy, elastic energy and a randomly perturbed atomic deposition flux. A linear stability analysis for the resulting sixth-order semilinear evolution equation for the thin film surface shows that that the new model allows for large anisotropy and gives rise to the formation of anisotropic quantum dots. The nonlinear three-dimensional evolution is investigated via numerical solutions. These suggest that increasing anisotropy stabilizes the faceted surfaces and may lead to a dramatic slowdown of the coarsening of the dots.