Mathematical Sciences: Dynamics in Partial Differential Equations Modeling Phase Transitions

[902422] Pego这个项目的主要目标是发展一种适合于承认相变的连续体模型的非线性动力学的数学理解。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？接口和转换区是如何传播的？具体而言，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定律的有效性；用正则化Lin模型和几何方法形式化地确定了向列液晶中奇点的迁移规律；可压缩流体中激波容许准则的发现，解释了亚稳性和均匀成核，以及最近在实验中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.