فهرست مطالب
Introduction
Contents
1 The Solution Theory for a Basic Class of Evolutionary Equations
1.1 The Time Derivative
1.2 A Hilbert Space Perspective on Ordinary Differential Equations
1.3 Evolutionary Equations
1.3.1 The Problem Class
1.3.2 The Solution Theory for Simple Material Laws
1.3.3 Lipschitz Continuous Perturbations
2 Some Applications to Models from Physics and Engineering
2.1 Acoustic Equations and Related Problems
2.1.1 The Classical Heat Equation
2.1.2 The Maxwell–Cattaneo-Vernotte Model
2.2 A Reduction Mechanism and the Relativistic Schrödinger Equation
2.2.1 Unitary Congruent Evolutionary Problems
2.2.2 The Relativistic Schrödinger Equation
2.3 Linear Elasticity
2.3.1 General (Non-symmetric) Linear(ized) Elasticity
2.3.2 The Isotropic Case
2.3.3 Symmetric Stresses
2.3.4 Linearized Incompressible Stokes Equations
2.4 The Guyer–Krumhansl Model of Thermodynamics
2.4.1 The Spatial Operator of the Guyer–Krumhansl Model
2.4.2 The Guyer–Krumhansl Model
2.5 The Equations of Electrodynamics
2.5.1 The Maxwell System as a Descendant of Elasticity
2.5.2 Non-classical Materials
2.5.3 Some Decomposition Results
2.5.4 The Extended Maxwell System
2.6 Coupled Physical Phenomena
2.6.1 The Coupling Recipe
2.6.2 The Propagation of Cavities
2.6.3 A Degenerate Reissner–Mindlin Plate Equation
2.6.4 Thermo-Piezo-Electro-Magnetism
3 But What About the Main Stream?
3.1 Where is the Laplacian?
3.2 Why Not Use Semi-Groups?
3.3 What About Other Types of Equations?
3.4 What About Other Boundary Conditions?
3.5 Why All This Functional Analysis?
A Two Supplements for the Toolbox
A.1 Mothers and Their Descendants
A.2 Abstract grad-div-Systems
B Requisites from Functional Analysis
B.1 Fundamentals of Hilbert Space Theory
B.2 The Projection Theorem
B.3 The Riesz Representation Theorem
B.4 Linear Operators and Their Adjoints
B.5 Duals and Adjoints
B.6 Solution Theory for (Real) Strictly Positive Linear Operators
B.7 An Approximation Result
B.8 The Root of Selfadjoint Accretive Operators and the Polar Decomposition
Bibliography
Index
Contents
1 The Solution Theory for a Basic Class of Evolutionary Equations
1.1 The Time Derivative
1.2 A Hilbert Space Perspective on Ordinary Differential Equations
1.3 Evolutionary Equations
1.3.1 The Problem Class
1.3.2 The Solution Theory for Simple Material Laws
1.3.3 Lipschitz Continuous Perturbations
2 Some Applications to Models from Physics and Engineering
2.1 Acoustic Equations and Related Problems
2.1.1 The Classical Heat Equation
2.1.2 The Maxwell–Cattaneo-Vernotte Model
2.2 A Reduction Mechanism and the Relativistic Schrödinger Equation
2.2.1 Unitary Congruent Evolutionary Problems
2.2.2 The Relativistic Schrödinger Equation
2.3 Linear Elasticity
2.3.1 General (Non-symmetric) Linear(ized) Elasticity
2.3.2 The Isotropic Case
2.3.3 Symmetric Stresses
2.3.4 Linearized Incompressible Stokes Equations
2.4 The Guyer–Krumhansl Model of Thermodynamics
2.4.1 The Spatial Operator of the Guyer–Krumhansl Model
2.4.2 The Guyer–Krumhansl Model
2.5 The Equations of Electrodynamics
2.5.1 The Maxwell System as a Descendant of Elasticity
2.5.2 Non-classical Materials
2.5.3 Some Decomposition Results
2.5.4 The Extended Maxwell System
2.6 Coupled Physical Phenomena
2.6.1 The Coupling Recipe
2.6.2 The Propagation of Cavities
2.6.3 A Degenerate Reissner–Mindlin Plate Equation
2.6.4 Thermo-Piezo-Electro-Magnetism
3 But What About the Main Stream?
3.1 Where is the Laplacian?
3.2 Why Not Use Semi-Groups?
3.3 What About Other Types of Equations?
3.4 What About Other Boundary Conditions?
3.5 Why All This Functional Analysis?
A Two Supplements for the Toolbox
A.1 Mothers and Their Descendants
A.2 Abstract grad-div-Systems
B Requisites from Functional Analysis
B.1 Fundamentals of Hilbert Space Theory
B.2 The Projection Theorem
B.3 The Riesz Representation Theorem
B.4 Linear Operators and Their Adjoints
B.5 Duals and Adjoints
B.6 Solution Theory for (Real) Strictly Positive Linear Operators
B.7 An Approximation Result
B.8 The Root of Selfadjoint Accretive Operators and the Polar Decomposition
Bibliography
Index
