Models of Phase Transitions

Reader's Guide.- 1. Some Nonlinear P.D.E.s.- Prelude.- I. Models and P.D.E.s.- 1. Modelling and Analysis.- 2. Nonlinear P.D.E.s and Minimization Problems.- 3. Examples of Nonlinear P.D.E.s.- 4. Comments.- II. A Class of Quasilinear Parabolic P.D.E.s.- 1. Variational Techniques of L2-Type.- 2. Further Results via L2-Techniques.- 3. Techniques of L1- and L ?-Type.- 4. Local Regularity Results.- 5. Integral Transformations.- 6. Semigroup Techniques.- 7. Comments.- III. Doubly Nonlinear Parabolic P.D.E.s.- 1. Doubly Nonlinear Parabolic Equations of First Type.- 2. Doubly Nonlinear Parabolic Equations of Second Type.- 3. Other Nonlinear Parabolic Equations.- 4. Use of Compactness by Strict Convexity.- 5. Comments.- 2. Phase Transitions.- IV. The Stefan Problem.- 1. Strong Formulation of the Stefan Problem.- 2. Surface Tension.- 3. Length Scales and Mushy Region.- 4. Weak Formulation of the Stefan Problem.- 5. On the Analysis of the Stefan Problem.- 6. Comparison between Strong and Weak Formulations.- 7. The Muskat and Hele-Shaw Problems.- 8. A Stefan-Type Problem Arising in Ferromagnetism.- 9. On the History of the Stefan Problem.- 10. Comments.- V. Generalizations of the Stefan Problem.- 1. Kinetic Undercooling and Phase Relaxation.- 2. Phase Transition in Two-Component Systems.- 3. Approach via Nonequilibrium Thermodynamics.- 4. Analysis of the Model of Section V.3.- 5. General Nonequilibrium Thermodynamics.- 6. The Evolution of the Free Energy.- 7. Comments.- VI. The Gibbs-Thomson Law.- 1. Free Energy.- 2. Entropy.- 3. Phase-Dependent Conductivity.- 4. The Gibbs-Thomson Law.- 5. The Phase Field Model.- 6. Comments.- VII. Nucleation and Growth.- 1. Local and Global Minimizers.- 2. Nucleation.- 3. Stable and Metastable States.- 4. Pure Phases.- 5. From Nucleation to Growth.- 6. Mean Curvature Flow.- 7. Nonlinear Mean Curvature Flow.- 8. Hysteresis in Front Motion.- 9. Comments.- VIII. The Stefan-Gibbs-Thomson Problem with Nucleation.- 1. Modes of Phase Transition.- 2. Formulation of the Problem.- 3. Some Auxiliary Results.- 4. Existence Result.- 5. The Mullins-Sekerka Problem.- 6. Comments.- IX. Two-Scale Models of Phase Transitions.- 1. Two-Scale Stefan Problem and Nonadiabatic Nucleation.- 2. Another Model with Surface Tension.- 3. A Mean Field Model.- 4. Micromagnetics.- 5. Some Comparisons.- 6. Comments.- X. Compactness by Strict Convexity.- 1. Extremality and Compactness.- 2. Strictly Convex Functionals.- 3. Applications.- 4. Comments.- XI. Toolbox.- 1. Some Function Spaces.- 2. Sobolev Spaces.- 3. Compactness.- 4. Convexity.- 5. Monotonicity.- 6. Accretiveness.- 7. Minimization.- 8. Geometric Measure Theory.- 9. Other Results.- Book Selection.