Analysis of a two-scale phase field model for liquid-solid phase transitions with equiaxed dendritic microstructure

We study a two-scale phase field model for liquid-solid phase transitions with equiaxed dendritic microstructure in binary mixtures. The model consists of a macroscopic heat equation and microscopic problems that describe the evolution of single equiaxed crystals. It is the formal homogenization limit of a phase field model under the assumption of a periodic initial distribution of equiaxed solid kernels for a regime with fast heat diffusion and slow solute diffusion. The existence, uniqueness, and a partial regularity of the solution is proved. These results are important for the justification of the formal homogenization by which it is derived.