This book presents a concise introduction to a unified Hilbert space approach to the mathematical modelling of physical phenomena which has been developed over recent years by Picard and his co-workers. The main focus is on time-dependent partial differential equations with a particular structure in the Hilbert space setting that ensures well-posedness and causality, two essential properties of any reasonable model in mathematical physics or engineering.However, the application of the theory to other types of equations is also demonstrated. By means of illustrative examples, from the straightforward to the more complex, the authors show that many of the classical models in mathematical physics as well as more recent models of novel materials and interactions are covered, or can be restructured to be covered, by this unified Hilbert space approach.
The reader should require only a basic foundation in the theory of Hilbert spaces and operators therein. For convenience, however, some of the more technical background requirements are covered in detail in two appendices The theory is kept as elementary as possible, making the material suitable for a senior undergraduate or master’s level course. In addition, researchers in a variety of fields whose work involves partial differential equations and applied operator theory will also greatly benefit from this approach to structuring their mathematical models in order that the general theory can be applied to ensure the essential properties of well-posedness and causality.
Author(s): Des McGhee, Rainer Picard, Sascha Trostorff, Marcus Waurick
Series: Frontiers in Mathematics
Edition: 1
Publisher: Birkhäuser
Year: 2020
Language: English
Pages: 183
Tags: Hilbert Spaces, Evolutionary Equations, Functional Analysis
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
