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.