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

9703530 Caginalp这项提议由数学科学司和材料研究司联合资助。这一建议涉及两个主题的研究：(A)界面的研究，以及(B)使用重整化和标度方法来研究抛物型微分方程组。对界面的研究沿着利用相场模型的先前工作的思路进行。就溶质在固体中的“冻结”而言，合金的研究尤其重要。这是一个数学可以做出重大贡献的领域，因为模型必然涉及退化的微分方程，因为固相中的扩散系数接近于零。在研究的第二部分，PI将利用重整化和尺度方法来获得抛物型微分方程组的结果，其中可以期望渐近自相似。这项工作涉及PI的工作的扩展，该工作使用这些方法来计算非线性扩散方程中的反常指数。其目的是将这些结果推广到抛物型微分方程组，从而为这些微分方程组的方法论增加一个强有力的工具。最后，将这些方法应用于反应扩散系统，如相场方程，以提取全局行为的关键特征。这项研究将为研究涉及材料科学的复杂问题提供一种形式化的方法。特别是，接口问题出现在许多工业应用中，并在理论和大规模计算方面构成了重要的挑战。发展一套一致的方程和高速计算方法是许多工业应用(如合金铸造)的起点。这项研究还将利用重整化技术，这些技术在理解热力学中的微妙行为方面非常成功。随着这些技术在这种动态环境中的发展，一个极其复杂的工程问题可能会被理解为具有特定特征行为的可管理部件。

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.