Phase Field Equations; Renormalization and Scaling in Differential Equations
Prof. Caginalp and collaborators developed many aspects of the phase field equations that describe interfaces using a smooth transition. A large number of free boundary problems including the classical Stefan model, the surface tension and kinetics model and Cahn-Hilliard have been shown to be distinguished limits of the phase field equations. The method has also been used to derive the limiting equations for alloy solidification. Recently, Chen, Eck and Caginalp have proposed a new phase field model that they proved differs from the sharp interface problem at only the second order (in interface thickness) leading to computations that are highly accurate.
Prof. Caginalp's initial paper on the phase field equations is the second most cited paper in the Archives RMA during the 20 year period 1984-2004.
The renormalization and scaling research focuses on techniques to calculate the exponents associated with large time and space behavior for the heat equation with nonlinear source terms, for example. Results have also been obtained for large time behavior of a solidification interface.
Prof. Caginalp also works in the area of Mathematical Finance and Economics (see publications under that heading).