Mathematical Sciences: Interface Dynamics and Renormalization Methods for Nonlinear Systems of Equations

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9703530 Caginalp This proposal is being jointly funded by Division of Mathematical Sciences (DMS) and Division of Materials Research (DMR). This proposal involves research on two topics: (a) the study of interfaces, and (b) the use of renormalization and scaling methods to study systems of parabolic differential equations. The study of interfaces proceeds along the lines of previous work that utilizes the phase field model. The study of alloys is particularly important in terms of the ''freezing in'' of solute into the solid. This is an area in which mathematics can make a substantial contribution since the models necessarily involve differential equations that are degenerate, as the diffusivity is close to zero in the solid phase. In the second part of the study, the PI will utilize renormalization and scaling methods to obtaing results on systems of parabolic differential equations in which an asymptotic self-similarity can be expected. The work involves an extension of the PI's work that used these methods to compute anomalous exponents in nonlinear diffusion equations. The objective is to extend these results to systems of parabolic differential equations, thereby adding a powerful tool to the methodology of these systems. Finally, the study will apply these methods to reaction-diffusion systems such as the phase field equations in order to extract key features of global behavior. This study will provide a formalism for studying complicated problems involving materials science. In particular, interface problems arise in many industrial applications and pose important challenges in terms of theory and large scale computation. The development of a consistent set of equations and methodology for high speed computation is valuable as a starting point for many industrial applications such as casting of alloys. The study will also utilize renormalization techniques that have been so successful in understanding subtle behavior in thermodynamics. With the development of these techniques in this dynamical context, an extremely complicated engineering problem can potentially be understood in terms of manageable parts with particular characteristic behavior.