WIAS Preprint No. 1582, (2010)

On a higher order convective Cahn--Hilliard type equation


  • Korzec, Maciek D.
  • Rybka, Piotr

2010 Mathematics Subject Classification

  • 35A05 74K35 35G25


  • Existence theory, global weak solutions, Galerkin approach, uniqueness, small-slope approximation, anisotropic surface energy, coarsening


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 $L^2(0,T; dot H^3_per)$. 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.

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