We study the coarsening of solutions of two models of multicomponent phase separation. One is a constant mobility system; the other is a degenerate mobility system. These models are natural generalizations of the Cahn-Hilliard equation to the case of a vector-valued order parameter. It has been conjectured that the characteristic length scale grows like as for the first case and for the second case. We prove a weak one-sided version of this assertion. Our method follows a strategy introduced by Kohn and Otto for problems with a scalar-valued order parameter; it combines a dissipation relationship with an isoperimetric inequality and an ODE argument. We also address a related model for anisotropic epitaxial growth.
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Robert V. Kohn, Xiaodong Yan, Coarsening rates for models of multicomponent phase separation. Interfaces Free Bound. 6 (2004), no. 1, pp. 135–149DOI 10.4171/IFB/94