Mathematical Sciences: Parallel and Distributed Diffusion Models for Heterogeneous Media

非均匀介质中扩散的数学模型将发展为非线性偏微分方程系统。这些扩散过程包括通过多孔或裂隙介质（具有饱和、吸收或相变）扩散和通过复合半导体材料传导的例子。这些体系的奇异性或简并性是由材料性质和组成物的几何形状决定的。主要目标包括通过双尺度模型的均匀化理论的发展，在适当的函数空间中证明适定性结果，以及对各种系统解的性质的研究。将发展复合介质中扩散的数学模型。这些扩散过程包括热传导（固-液相变）、流体通过多孔或裂隙介质的扩散（饱和）、颗粒状物质的吸收以及半导体材料中电流的宏观描述。这些体系的奇异性或简并性源于材料的性质和组成物的几何形状。主要目标包括发展精确的小尺度模型来描述介质组分之间复杂界面的流动，以及计算解所必需的平均大尺度模型。

DMS-9500920 Showalter Mathematical models of diffusion in heterogeneous media will be developed as nonlinear systems of partial differential equations. These diffusion processes include examples of diffusion through porous or fissured media (with saturation, absorption, or phase change) and conduction through composite semiconductor materials. The singular or degenerate nature of these systems arise from material properties and from the geometry of the composition. Major objectives include the development by homogenization theory of two-scale models, proof of well-posedness results in appropriate function spaces, and the investigation of properties of solutions for the various systems. Mathematical models of diffusion in composite media will be developed. These diffusion processes include examples of heat conduction (with solid - liquid phase change), diffusion of fluid through porous or fissured media (with saturation), absorption by granulated substance, and the macroscopic description of current flow in semi-conductor materials. The singular or degenerate nature of these systems arise from properties of the materials and from the geometry of the composition. Major objectives include the development of exact small-scale models to describe the flow across the intricate interface between the components of the media and averaged large-scale models which are necessary for the computation of solutions.