Models of Phase Transitions

ISBN-13: 9781461286417 / Angielski / Miękka / 2011 / 326 str.

Augusto Visintin

Opis

... "What do you call work?" "Why ain't that work?" Tom resumed his whitewashing, and answered carelessly: "Well. lI1a), he it is, and maybe it aill't. All I know, is, it suits Tom Sawvc/: " "Oil CO/lll , IIOW, Will do not mean to let 011 that you like it?" The brush continued to move. "Likc it? Well, I do not see wlzy I oughtn't to like it. Does a hoy get a chance to whitewash a fence every day?" That put the thing ill a Ilew light. Ben stopped nibhling the apple .... (From Mark Twain's Adventures of Tom Sawyer, Chapter II.) Mathematics can put quantitative phenomena in a new light; in turn applications may provide a vivid support for mathematical concepts. This volume illustrates some aspects of the mathematical treatment of phase transitions, namely, the classical Stefan problem and its generalizations. The in tended reader is a researcher in application-oriented mathematics. An effort has been made to make a part of the book accessible to beginners, as well as physicists and engineers with a mathematical background. Some room has also been devoted to illustrate analytical tools. This volume deals with research I initiated when I was affiliated with the Istituto di Analisi Numerica del C.N.R. in Pavia, and then continued at the Dipartimento di Matematica dell'Universita di Trento. It was typeset by the author in plain TEX.

Recenzje

"The book is well organized, concise and clearly written with a strict interplay between physics and mathematics... Largely self-contained ... highly recommended to all graduate students and reserachers in applied mathematics."

--ZAA

Spis treści

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.