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Beyond Partial Differential Equations

On Linear and Quasi-Linear Abstract Hyperbolic Evolution Equations

  • Book
  • © 2007

Overview

Authors:
  1. Horst Reinhard Beyer

    1. Louisiana State University, Baton Rouge, USA

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Part of the book series: Lecture Notes in Mathematics (LNM, volume 1898)

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Table of contents (12 chapters)

  1. Front Matter

    Pages i-xiv
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  2. Conventions

    Pages 1-3

  3. Mathematical Introduction

    Pages 5-12

  4. Prerequisites

    Pages 13-39

  5. Strongly Continuous Semigroups

    Pages 41-69

  6. Examples of Generators of Strongly Continuous Semigroups

    Pages 71-103

  7. Intertwining Relations, Operator Homomorphisms

    Pages 105-121

  8. Examples of Constrained Systems

    Pages 123-135

  9. Kernels, Chains, and Evolution Operators

    Pages 137-163

  10. The Linear Evolution Equation

    Pages 165-176

  11. Examples of Linear Evolution Equations

    Pages 177-214

  12. The Quasi-Linear Evolution Equation

    Pages 215-234

  13. Examples of Quasi-Linear Evolution Equations

    Pages 235-263

  14. Back Matter

    Pages 265-287
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Keywords

About this book

Semigroup Theory uses abstract methods of Operator Theory to treat initial bou- ary value problems for linear and nonlinear equations that describe the evolution of a system. Due to the generality of its methods, the class of systems that can be treated in this way exceeds by far those described by equations containing only local op- ators induced by partial derivatives, i.e., PDEs. In particular, that class includes the systems of Quantum Theory. Another important application of semigroup methods is in ?eld quantization. Simple examples are given by the cases of free ?elds in Minkowski spacetime like Klein-Gordon ?elds, the Dirac ?eld and the Maxwell ?eld, whose ?eld equations are given by systems of linear PDEs. The second quantization of such a ?eld replaces the ?eld equation by a Schrodinger ¨ equation whose Hamilton operator is given by the second quantization of a non-local function of a self-adjoint linear operator. That operator generates the semigroup given by the time-development of the solutions of the ?eld equation corresponding to arbitrary initial data as a function of time.

Reviews

From the reviews:

"In the monograph under review, designed as a textbook for graduate students, the author aims to build up the theory in a straightforward way in view of physical applications. … The material covered in the book was used in a two-semester course for graduate students. The book is a good summary of the most important facts of the theory of evolution equations, and is also a good source of information for researchers looking for new applications of this theory." (András Bátkai, Mathematical Reviews, Issue 2008 h)

"This well-written work is the outgrowth of a two-semester course on linear semigroup methods, linear and quasi-linear hyperbolic systems of partial differential equations, and the abstract theory of evolution equations … . Although the text was written for truly advanced graduate students, it contains a wealth of well presented results on semigroup theory of evolution equations … and will therefore serve as a valuable resource for researchers in mathematics and theoretical physics as well." (Thomas Hagen, Zentralblatt MATH, Vol. 1144, 2008)

Authors and Affiliations

  • Louisiana State University, Baton Rouge, USA

    Horst Reinhard Beyer

Bibliographic Information

  • Book Title: Beyond Partial Differential Equations

  • Book Subtitle: On Linear and Quasi-Linear Abstract Hyperbolic Evolution Equations

  • Authors: Horst Reinhard Beyer

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-540-71129-2

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2007

  • Softcover ISBN: 978-3-540-71128-5Published: 04 April 2007

  • eBook ISBN: 978-3-540-71129-2Published: 10 April 2007

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: XIV, 283

  • Topics: Analysis, Operator Theory, Partial Differential Equations

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