Evolutionary Equations : Picard's Theorem for Partial Differential Equations, and Applications.
- 1st ed.
- 1 online resource (321 pages)
- Operator Theory: Advances and Applications Series ; v.287 .
- Operator Theory: Advances and Applications Series .
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-Lindel�of 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.