The study of phase transitions is one of the fundamental problems of physics. The goal of this seminar was to understand better the spectacular progress made recently in constructing continuum models. Concentrating on a few examples such as the microstructure of crystals, defects in liquid crystals and liquid-vapor interfaces, several key points are described, for example the structure and evolution of the interfaces, regularization via interfacial energy, and equilibrium theories. The mathematical treatment of these questions involves large-oscillation theories (Young's measures, compensated compactness), spectral theory, admissibility of shock waves, long-time behavior of dynamical systems, high-order perturbations, group actions, solitons, and others.
Table of ContentsClusters of singularities in liquid crystals.- Stability analysis of flows of liquid crystal polymers exhibiting a banding structure.- Coupled transverse-longitudinal solitons in elastic crystals.- The computation of the austenitic-martensitic phase transition.- Theory of diffusionless phase transitions.- Invariance properties of inviscid fluids of grade n.- On diffusion in two-phase systems: the sharp interface versus the transition layer.- Instabilities in shear flow of viscoelastic fluids with fading memory.- Singularities of the order parameter in condensed matter physics.- Swelling and shrinking of polyelectrolyte gels.- Adiabatic phase changes, fast and slow.- The continuous structure of discontinuities.- Riemann problems involving undercompressive shocks.- The viscosity-capillarity approach to phase transitions.- A version of the fundamental theorem for young measures.- Very slow phase separation in one dimension.