Coarsening rates for models of multicomponent phase separation (Q1430456)

Summary: 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 \(\ell(t)\) grows like \(t^{1/3}\) as \(t\to\infty\) for the first case and \(\ell\sim t^{1/4}\) for the second case. We prove a weak one-sided version of this assertion. Our method follows a strategy introduced by \textit{R. V. Kohn} and \textit{F. Otto} [Commun. Math. Phys. 229, 375--395 (2002; Zbl 1004.82011)] 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.