Coarsening rates for models of multicomponent phase separation

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 `(t) grows like t1/3 as t →∞ for the first case and ` ∼ t1/4 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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