 
Summary: SECONDORDER PHASE FIELD ASYMPTOTICS
FOR UNEQUAL CONDUCTIVITIES
ROBERT F. ALMGREN
SIAM J. APPL. MATH. c 1999 Society for Industrial and Applied Mathematics
Vol. 59, No. 6, pp. 20862107
Abstract. We extend Karma and Rappel's improved asymptotic analysis of the phase field
model to different diffusivities in solid and liquid. We consider both secondorder "classical" asymp
totics, in which the interface thickness is taken much smaller than the capillary length, and the new
"isothermal" asymptotics, in which the two lengths are considered comparable. In the first case,
if the phase field model is required to be gradient flow for an entropy functional, then for unequal
diffusivities it is impossible to construct a phase equation with finite kinetics which converges with
secondorder accuracy to a GibbsThomson equilibrium condition with infinitely fast kinetics. In the
second case, some error terms are pushed to higher orders, and it is easy to eliminate the remaining
errors with finite phase kinetics.
Key words. phase field asymptotics, diffusivity
AMS subject classifications. 80A22, 35K57, 35R35, 41A60
PII. S0036139997330027
1. Introduction. We consider the motion of an interface (t), either a curve
in two dimensions, or a surface in three dimensions, which divides a fixed enclosing
domain into two bulk regions +(t) and (t). For solidification, + is the liquid
