Mathematical Sciences: Dynamics in Partial Differential Equations Modeling Phase Transitions

8902422 PEGO这个项目的主要目标是发展对非线性动力学的数学理解，适用于允许相变的连续介质模型。要研究的数学问题涉及一般问题：非线性系统如何松弛到平衡？界面和过渡区如何传播？具体地，该项目包括研究Cahn-Hilliard方程和相场方程中过渡层的传播，通过奇异摄动的几何方法证明界面迁移规律的有效性，使用LIN的正则化模型和几何方法形式化地确定向列相液晶中奇点的迁移规律；可压缩流体中冲击波的可容许性准则的发现解释了亚稳和均匀成核，这是最近在实验中被证明高度相关的非平衡现象。

8902422 Pego The main goal of this project is to develop a mathematical understanding of nonlinear dynamics appropriate for models of continua which admit phase transitions. The mathematical problems to be studied address the general questions: How do nonlinear systems relax to equilibrium? How do interfaces and transition zones propagate? Specifically, the project includes the study of the propagation of transition layers in the Cahn-Hilliard equation and phase field equations, to prove the validity of interface migration laws via the geometric approach to singular perturbations; the formal determination of the laws of migration for singularities in nematic liquid crystals using a regularized model of Lin and the geometric method; the discovery of admissibility criteria for shock waves in compressible fluids which account for metastability and homogeneous nucleation, nonequilibrium phenomena which have recently been shown to be highly relevant in experiment.