Convergence of the phase field model to its sharp interface limits

We consider the distinguished limits of the phase field equations and prove that the corresponding free boundary problem is attained in each case. These include the classical Stefan model, the surface tension model (with or without kinetics), the surface tension model with zero specific heat, the two phase Hele-Shaw, or quasi-static, model. The Hele-Shaw model is also a limit of the Cahn-Hilliard equation, which is itself a limit of the phase field equations. Also included in the distinguished limits is the motion by mean curvature model that is a limit of the Allen-Cahn equation, which can in turn be attained from the phase field equations.