Seifert C., Trostorff S., Waurick M. Evolutionary Equations: Picard's Theorem for Partial Differential Equations, and Applications

Basel: Birkhäuser, 2022. — 321 p.This open access book provides a solution theory for time-dependent partial differential equations, which classically have not been accessible by a unified method. Instead of using sophisticated techniques and methods, the approach is elementary in the sense that only Hilbert space methods and some basic theory of complex analysis are required. Nevertheless, key properties of solutions can be recovered in an elegant manner. Moreover, the strength of this method is demonstrated by a large variety of examples, showing the applicability of the approach of evolutionary equations in various fields. Additionally, a quantitative theory for evolutionary equations is developed.
The text is self-contained, providing an excellent source for a first study on evolutionary equations and a decent guide to the available literature on this subject, thus bridging the gap to state-of-the-art mathematical research.From ODEs to PDEs
Time-independent Problems
Evolutionary Equations
Particular Examples and the Change of Perspective
A Brief Outline of the Course
Comments
ExercisesUnbounded Operators
Operators in Banach Spaces
Operators in Hilbert Spaces
Computing the Adjoint
The Spectrum and Resolvent Set
Comments
ExercisesThe Time Derivative
Bochner–Lebesgue Spaces
The Time Derivative as a Normal Operator
Comments
ExercisesOrdinary Differential Equations
The Domain of the time derivative and the Sobolev Embedding Theorem
The Picard–Lindelöf Theorem
Delay Differential Equations
Comments
ExercisesThe Fourier–Laplace Transformation and Material Law Operators
The Fourier Transformation
The Fourier–Laplace Transformation and Its Relation to the Time Derivative
Material Law Operators
Comments
ExercisesSolution Theory for Evolutionary Equations
First Order Sobolev Spaces
Well-Posedness of Evolutionary Equations and Applications
Proof of Picard's Theorem
Comments
ExercisesExamples of Evolutionary Equations
Poro-Elastic Deformations
Fractional Elasticity
The Heat Equation with Delay
Dual Phase Lag Heat Conduction
Comments
ExercisesCausality and a Theorem of Paley and Wiener
A Theorem of Paley and Wiener
A Representation Result
Comments
ExercisesInitial Value Problems and Extrapolation Spaces
What are Initial Values?
Extrapolating Operators
Evolutionary Equations in Distribution Spaces
Initial Value Problems for Evolutionary Equations
Comments
ExercisesDifferential Algebraic Equations
The Finite-Dimensional Case
The Infinite-Dimensional Case
Comments
ExercisesExponential Stability of Evolutionary Equations
The Notion of Exponential Stability
A Criterion for Exponential Stability of Parabolic-Type Equations
Three Exponentially Stable Models for Heat Conduction
Exponential Stability for Hyperbolic-Type Equations
A Criterion for Exponential Stability of Hyperbolic-Type Equations
Examples of Exponentially Stable Hyperbolic Problems
Comments
ExercisesBoundary Value Problems and Boundary Value Spaces
The Boundary Values of Functions in the Domain of the Gradient
The Boundary Values of Functions in the Domain of the Divergence
Inhomogeneous Boundary Value Problems
Abstract Boundary Data Spaces
Robin Boundary Conditions
Comments
ExercisesContinuous Dependence on the Coefficients I
Convergence of Material Laws
A Leading Example
Convergence in the Weak Operator Topology
Comments
ExercisesContinuous Dependence on the Coefficients II
A Convergence Theorem
The Theorem of Rellich and Kondrachov
The Periodic Gradient
The Limit of the Scaled Coefficient Sequence
Comments
ExercisesMaximal Regularity
Guiding Examples and Non-Examples
The Maximal Regularity Theorem and Fractional Sobolev Spaces
The Proof of Theorem 15.2.3
Comments
ExercisesNon-Autonomous Evolutionary Equations
Examples
Non-Autonomous Picard's Theorem—The ODE Case
Non-Autonomous Picard's Theorem—The PDE Case
Comments
ExercisesEvolutionary Inclusions
Maximal Monotone Relations and the Theorem of Minty
The Yosida Approximation and Perturbation Results
A Solution Theory for Evolutionary Inclusions
Maxwell's Equations in Polarisable Media
Comments
ExercisesDerivations of Main Equations
Heat Equation
Maxwell's Equations
Linear Elasticity
Scalar Wave Equation
Comments
Exercises