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E-raamat: Beyond Partial Differential Equations: On Linear and Quasi-Linear Abstract Hyperbolic Evolution Equations

  • Formaat: PDF+DRM
  • Sari: Lecture Notes in Mathematics 1898
  • Ilmumisaeg: 10-Apr-2007
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Keel: eng
  • ISBN-13: 9783540711292
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  • Formaat: PDF+DRM
  • Sari: Lecture Notes in Mathematics 1898
  • Ilmumisaeg: 10-Apr-2007
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Keel: eng
  • ISBN-13: 9783540711292

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The present volume is self-contained and introduces to the treatment of linear and nonlinear (quasi-linear) abstract evolution equations by methods from the theory of strongly continuous semigroups. The theoretical part is accessible to graduate students with basic knowledge in functional analysis. Only some examples require more specialized knowledge from the spectral theory of linear, self-adjoint operators in Hilbert spaces. Particular stress is on equations of the hyperbolic type since considerably less often treated in the literature. Also, evolution equations from fundamental physics need to be compatible with the theory of special relativity and therefore are of hyperbolic type. Throughout, detailed applications are given to hyperbolic partial differential equations occurring in problems of current theoretical physics, in particular to Hermitian hyperbolic systems. This volume is thus also of interest to readers from theoretical physics.

This book introduces the treatment of linear and nonlinear (quasi-linear) abstract evolution equations by methods from the theory of strongly continuous semigroups. The theoretical part is accessible to graduate students with basic knowledge in functional analysis, with only some examples requiring more specialized knowledge from the spectral theory of linear, self-adjoint operators in Hilbert spaces. Emphasis is placed on equations of the hyperbolic type which are less often treated in the literature.

Arvustused

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)

Conventions
1(5)
Mathematical Introduction
5(8)
Quantum Theory
5(3)
Wave Equations
8(5)
Prerequisites
13(28)
Linear Operators in Banach Spaces
13(12)
Weak Integration of Banach Space-Valued Maps
25(10)
Exponentials of Bounded Linear Operators
35(6)
Strongly Continuous Semigroups
41(30)
Elementary Properties
42(9)
Characterizations
51(7)
An Integral Representation in the Complex Case
58(1)
Perturbation Theorems
59(4)
Strongly Continuous Groups
63(3)
Associated Inhomogeneous Initial Value Problems
66(5)
Examples of Generators of Strongly Continuous Semigroups
71(34)
The Ordinary Derivative on a Bounded Interval
71(3)
Linear Stability of Ideal Rotating Couette Flows
74(3)
Outgoing Boundary Conditions
77(7)
Damped Wave Equations
84(13)
Autonomous Linear Hermitian Hyperbolic Systems
97(8)
Intertwining Relations, Operator Homomorphisms
105(18)
Semigroups and Their Restrictions
105(9)
Intertwining Relations
114(3)
Nonexpansive Homomorphisms
117(6)
Examples of Constrained Systems
123(14)
1-D Wave Equations with Sommerfeld Boundary Conditions
123(4)
1-D Wave Equations with Engquist-Majda Boundary Conditions
127(5)
Maxwell's Equations in Flat Space
132(5)
Kernels, Chains, and Evolution Operators
137(28)
A Convolution Calculus with Operator-Valued Kernels
138(8)
Chains
146(1)
Juxtaposition of Chains
147(1)
Finitely Generated Chains
148(1)
Evolution Operators
149(6)
Stable Families of Generators
155(10)
The Linear Evolution Equation
165(12)
Examples of Linear Evolution Equations
177(38)
Scalar Fields in the Gravitational Field of a Spherical Black Hole
178(21)
Non-Autonomous Linear Hermitian Hyperbolic Systems
199(16)
The Quasi-Linear Evolution Equation
215(20)
Examples of Quasi-Linear Evolution Equations
235(30)
A Generalized Inviscid Burgers' Equation
235(11)
Quasi-Linear Hermitian Hyperbolic Systems
246(19)
Appendix
265(4)
References 269(10)
Index of Notation 279(2)
Index of Terminology 281