Dissipative Phase Transitions.
Title:
Dissipative Phase Transitions.
Author:
Colli, Pierluigi.
ISBN:
9789812774293
Personal Author:
Physical Description:
1 online resource (321 pages)
Series:
Series on Advances in Mathematics for Applied Sciences ; v.71
Series on Advances in Mathematics for Applied Sciences
Contents:
CONTENTS -- Preface -- Mathematical models including a hysteresis operator -- 1 Introduction -- 2 Mathematical treatment for hysteresis operator -- 2.1 Play operator -- 2.2 Stop operator -- 2.3 The Duhem model -- 3 Shape memory alloys -- 4 Examples of hysteresis operator -- 4.1 Solid-liquid phase transition -- 4.2 Biological model -- 4.3 Magnetostrictive thin film multi-layers -- References -- Modelling phase transitions via an entropy equation: long-time behaviour of the solutions -- 1 Introduction -- 2 The model and the resulting PDE's system -- 3 Main results -- 4 The existence and uniqueness result -- 4.1 Proof of Theorem 5 -- 5 Uniform estimates on (0. +oo) -- 6 The w-limit -- References -- Global solution to a one dimensional phase transition model with strong dissipation -- 1 Introduction and derivation of the model -- 2 Notation and main results -- 3 Proof of Theorem 1 -- 4 Proof of Theorem 2 -- References -- A global in time result for an integro-differential parabolic inverse problem in the space of bounded functions -- 1 Introduction -- 2 Definitions and main results -- 2.1 The main abstract result -- 2.2 An application -- 3 The weighted spaces -- 4 An equivalent fixed point system -- 5 Proof of Theorem 6 -- References -- Weak solutions for Stefan problems with convections -- 1 Introduction -- 2 Stefan problem in non-cylindrical domain with convection governed by Navier-Stokes equations -- 2.1 Classical formulation -- 2.2 Weak formulation and existence result -- 3 Transmission-Stefan problem -- 4 Mathematical modelling for the transmission problem with the Stefan and Navier-Stokes equations -- References -- Memory relaxation of the one-dimensional Cahn-Hilliard equation -- 1 Introduction -- 1.1 The model equation -- 1.2 The singular limit -- 1.3 The history space formulation.
1.4 Assumptions on the memoiy kernel and on the nonlinearity -- 2 The dynamical system -- 3 The main result -- 4 Sketch of the proof of Theorem 4 -- References -- Mathematical models for phase transition in materials with thermal memory -- 1 Introduction -- 2 Notations and basic assumptions -- 3 Phase-field models with a modified energy balance -- 4 A phase-field model with thermal memory -- 4.1 Thermodynamic restrictions -- 4.2 Evolution equations -- 5 Phase-field models with an entropy extra-fiux -- 5.1 Thermodynamic restrictions and evolution equations -- 5.2 Quasilinear theory -- 6 Appendix: quasilinear approximation -- References -- Hysteresis in a first order hyperbolic equation -- 1 Introduction -- 2 Hysteresis -- 3 Semigroup approach -- 4 Main result -- References -- Approximation of inverse problems related to parabolic integro-differential systems of Caginalp type -- 1 Introduction -- 2 The problem with exact data -- 2.1 The direct problem with exact data -- 2.2 The inverse problem with exact data -- 3 The inverse problem with approximate data -- 3.1 Well-posedness of the problem with approximate data -- 3.2 Uniform estimates of the solution to Problem (PQ) -- 4 Convergence estimates -- 5 Apppendix -- References -- Gradient flow reaction/diffusion models in phase transitions -- 1 Introduction -- 1.1 Structure and results -- 2 Canonical equations -- 3 Expansions -- 4 T-convergence - Steady problem -- 5 T-convergence - Time-dependent problem -- 6 Conclusion -- References -- New existence result for a 3-D shape memory model -- 1 Introduction -- 1.1 Notation -- 2 Assumptions and main results -- 3 Parabolicity of the elasticity system with viscosity and capillarity -- 4 Auxiliary existence results for parabolic problems of fourth and second order -- 5 Outline of the proof of Theorem 1 -- References.
Analysis of a 1-D thermoviscoelastic model with temperature-dependent viscosity -- 1 Introduction -- 2 Notation and assumptions -- 3 Continuous problem -- 4 Continuous dependence -- 5 Approximation -- 6 Discrete well-posedness -- 7 Stability -- 8 Convergence -- 9 Error control -- References -- Global attractor for the weak solutions of a class of viscous Cahn-Hilliard equations -- 1 Introduction -- 1.1 Plan of the paper -- 2 Preliminaries: generalized semiflows -- 2.1 Definition of generalized semiflow -- 2.2 Continuity properties -- 2.3 Compactness and dissipativity -- 2.4 Existence of the attractor -- 3 Main results -- 3.1 Notation -- 3.2 Assumptions on the data -- 3.3 Statement of the problem -- 3.4 Existence of weak solutions -- 3.5 Generalized semiflow and long-time behavior of the weak solutions -- 4 Generalized semiflow and Global Attractor of the weak solutions -- 4.1 Proof of Theorem 9 -- 4.2 Proof of Theorem 11 -- 4.3 Proof of Theorem 12 -- References -- Stability for phase field systems involving indefinite surface tension coefficients -- 1 Introduction -- 2 Preliminaries -- 3 Key properties for the system (S) -- 4 Steady-state patterns -- 5 Stability for steady-state patterns -- References -- Geometric features of p-Laplace phase transitions -- 1 Introduction -- 2 Results -- References.
Abstract:
Phase transition phenomena arise in a variety of relevant real world situations, such as melting and freezing in a solid-liquid system, evaporation, solid-solid phase transitions in shape memory alloys, combustion, crystal growth, damage in elastic materials, glass formation, phase transitions in polymers, and plasticity. The practical interest of such phenomenology is evident and has deeply influenced the technological development of our society, stimulating intense mathematical research in this area. This book analyzes and approximates some models and related partial differential equation problems that involve phase transitions in different contexts and include dissipation effects. Contents: Mathematical Models Including a Hysteresis Operator (T Aiki); Modelling Phase Transitions via an Entropy Equation: Long-Time Behavior of the Solutions (E Bonetti); Global Solution to a One Dimensional Phase Transition Model with Strong Dissipation (G Bonfanti & F Luterotti); A Global in Time Result for an Integro-Differential Parabolic Inverse Problem in the Space of Bounded Functions (F Colombo et al.); Weak Solutions for Stefan Problems with Convections (T Fukao); Memory Relaxation of the One-Dimensional Cahn-Hilliard Equation (S Gatti et al.); Mathematical Models for Phase Transition in Materials with Thermal Memory (G Gentili & C Giorgi); Hysteresis in a First Order Hyperbolic Equation (J Kopfová); Approximation of Inverse Problems Related to Parabolic Integro-Differential Systems of Caginalp Type (A Lorenzi & E Rocca); Gradient Flow Reaction/Diffusion Models in Phase Transitions (J Norbury & C Girardet); New Existence Result for a 3-D Shape Memory Model (I Pawlow & W M Zajaczkowski); Analysis of a 1-D Thermoviscoelastic Model with Temperature-Dependent Viscosity (R Peyroux & U Stefanelli); Global Attractor for the Weak Solutions of a Class of Viscous
Cahn-Hilliard Equations (R Rossi); Stability for Phase Field Systems Involving Indefinite Surface Tension Coefficients (K Shirakawa); Geometric Features of p -Laplace Phase Transitions (E Valdinoci). Readership: Applied mathematicians and researchers in analysis and differential equations.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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