Mathematical Sciences: Dynamics in Partial Differential Equations Modeling Phase Transitions

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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.