Front stability in mean-field models of diffusion-limited growth

We present calculations of the stability of planar fronts in two mean-field models of diffusion-limited growth. The steady state solution for the, front can exist for a continuous family of velocities, and we show that the selected velocity is given by marginal stability theory. We find that a naive mean-field theory has no instability to transverse perturbations, while a threshold mean-field theory has a Mullins-Sekerka instability. These results place on firm theoretical ground the observed lack of the dendritic morphology in naive mean-field theory and its presence in threshold models. The existence of a Mullins-Sekerka instability is related to the behavior of the mean-field theories in the zero-undercooling limit.