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Evolutionary Equations : Picard's Theorem for Partial Differential Equations, and Applications.

Bibliographic Details
Main Author: Seifert, Christian.
Other Authors: Trostorff, Sascha., Waurick, Marcus.
Format: eBook
Language:English
Published: Cham : Springer International Publishing AG, 2022.
Edition:1st ed.
Series:Operator Theory: Advances and Applications Series
Subjects:
Electronic books.
Online Access:Click to View
Table of Contents:
  • Intro
  • Preface
  • Contents
  • 1 Introduction
  • 1.1 From ODEs to PDEs
  • 1.2 Time-independent Problems
  • 1.3 Evolutionary Equations
  • 1.4 Particular Examples and the Change of Perspective
  • 1.5 A Brief Outline of the Course
  • 1.6 Comments
  • Exercises
  • References
  • 2 Unbounded Operators
  • 2.1 Operators in Banach Spaces
  • 2.2 Operators in Hilbert Spaces
  • 2.3 Computing the Adjoint
  • 2.4 The Spectrum and Resolvent Set
  • 2.5 Comments
  • Exercises
  • References
  • 3 The Time Derivative
  • 3.1 Bochner-Lebesgue Spaces
  • 3.2 The Time Derivative as a Normal Operator
  • 3.3 Comments
  • Exercises
  • References
  • 4 Ordinary Differential Equations
  • 4.1 The Domain of the time derivative and the Sobolev Embedding Theorem
  • 4.2 The Picard-Lindelöf Theorem
  • 4.3 Delay Differential Equations
  • 4.4 Comments
  • Exercises
  • References
  • 5 The Fourier-Laplace Transformation and Material Law Operators
  • 5.1 The Fourier Transformation
  • 5.2 The Fourier-Laplace Transformation and Its Relation to the Time Derivative
  • 5.3 Material Law Operators
  • 5.4 Comments
  • Exercises
  • References
  • 6 Solution Theory for Evolutionary Equations
  • 6.1 First Order Sobolev Spaces
  • 6.2 Well-Posedness of Evolutionary Equations and Applications
  • 6.3 Proof of Picard's Theorem
  • 6.4 Comments
  • Exercises
  • References
  • 7 Examples of Evolutionary Equations
  • 7.1 Poro-Elastic Deformations
  • 7.2 Fractional Elasticity
  • 7.3 The Heat Equation with Delay
  • 7.4 Dual Phase Lag Heat Conduction
  • 7.5 Comments
  • Exercises
  • References
  • 8 Causality and a Theorem of Paley and Wiener
  • 8.1 A Theorem of Paley and Wiener
  • 8.2 A Representation Result
  • 8.3 Comments
  • Exercises
  • References
  • 9 Initial Value Problems and Extrapolation Spaces
  • 9.1 What are Initial Values?
  • 9.2 Extrapolating Operators
  • 9.3 Evolutionary Equations in Distribution Spaces.
  • 9.4 Initial Value Problems for Evolutionary Equations
  • 9.5 Comments
  • Exercises
  • References
  • 10 Differential Algebraic Equations
  • 10.1 The Finite-Dimensional Case
  • 10.2 The Infinite-Dimensional Case
  • 10.3 Comments
  • Exercises
  • References
  • 11 Exponential Stability of Evolutionary Equations
  • 11.1 The Notion of Exponential Stability
  • 11.2 A Criterion for Exponential Stability of Parabolic-Type Equations
  • 11.3 Three Exponentially Stable Models for Heat Conduction
  • 11.4 Exponential Stability for Hyperbolic-Type Equations
  • 11.5 A Criterion for Exponential Stability of Hyperbolic-Type Equations
  • 11.6 Examples of Exponentially Stable Hyperbolic Problems
  • 11.7 Comments
  • Exercises
  • References
  • 12 Boundary Value Problems and Boundary Value Spaces
  • 12.1 The Boundary Values of Functions in the Domain of the Gradient
  • 12.2 The Boundary Values of Functions in the Domain of the Divergence
  • 12.3 Inhomogeneous Boundary Value Problems
  • 12.4 Abstract Boundary Data Spaces
  • 12.5 Robin Boundary Conditions
  • 12.6 Comments
  • Exercises
  • References
  • 13 Continuous Dependence on the Coefficients I
  • 13.1 Convergence of Material Laws
  • 13.2 A Leading Example
  • 13.3 Convergence in the Weak Operator Topology
  • 13.4 Comments
  • Exercises
  • References
  • 14 Continuous Dependence on the Coefficients II
  • 14.1 A Convergence Theorem
  • 14.2 The Theorem of Rellich and Kondrachov
  • 14.3 The Periodic Gradient
  • 14.4 The Limit of the Scaled Coefficient Sequence
  • 14.5 Comments
  • Exercises
  • References
  • 15 Maximal Regularity
  • 15.1 Guiding Examples and Non-Examples
  • 15.2 The Maximal Regularity Theorem and Fractional Sobolev Spaces
  • 15.3 The Proof of Theorem 15.2.3
  • 15.4 Comments
  • Exercises
  • References
  • 16 Non-Autonomous Evolutionary Equations
  • 16.1 Examples
  • 16.2 Non-Autonomous Picard's Theorem-The ODE Case.
  • 16.3 Non-Autonomous Picard's Theorem-The PDE Case
  • 16.4 Comments
  • Exercises
  • References
  • 17 Evolutionary Inclusions
  • 17.1 Maximal Monotone Relations and the Theorem of Minty
  • 17.2 The Yosida Approximation and Perturbation Results
  • 17.3 A Solution Theory for Evolutionary Inclusions
  • 17.4 Maxwell's Equations in Polarisable Media
  • 17.5 Comments
  • Exercises
  • References
  • A Derivations of Main Equations
  • A.1 Heat Equation
  • A.2 Maxwell's Equations
  • A.3 Linear Elasticity
  • A.4 Scalar Wave Equation
  • A.5 Comments
  • Exercises
  • References
  • Bibliography
  • Index.

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