We prove an upper bound on the coarsening rate for solutions of a phase field model with arbitrarily complicated patterns of phases. The analysis is performed in a regime corresponding to the late stages of phase separation, in which the ratio between the transition layer thickness and the length scale of the pattern is small, and is also small compared to the square of the ratio between the pattern scale and the system size. The analysis extends the method of Kohn and Otto (Comm.\ Math.\ Phys.\ 229 (2002), 375-395) to deal with both temperature and phase fields.
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Shibin Dai, Robert L. Pego, An upper bound on the coarsening rate for mushy zones in a phase-field model. Interfaces Free Bound. 7 (2005), no. 2, pp. 187–197DOI 10.4171/IFB/120