9781584882503

Abstract Cauchy Problems: Three Approaches

by ;
  • ISBN13:

    9781584882503

  • ISBN10:

    1584882506

  • Edition: 1st
  • Format: Hardcover
  • Copyright: 2001-03-27
  • Publisher: Chapman & Hall/

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Summary

Although the theory of well-posed Cauchy problems is reasonably understood, ill-posed problems-involved in a numerous mathematical models in physics, engineering, and finance- can be approached in a variety of ways. Historically, there have been three major strategies for dealing with such problems: semigroup, abstract distribution, and regularization methods. Semigroup and distribution methods restore well-posedness, in a modern weak sense. Regularization methods provide approximate solutions to ill-posed problems. Although these approaches were extensively developed over the last decades by many researchers, nowhere could one find a comprehensive treatment of all three approaches.Abstract Cauchy Problems: Three Approaches provides an innovative, self-contained account of these methods and, furthermore, demonstrates and studies some of the profound connections between them. The authors discuss the application of different methods not only to the Cauchy problem that is not well-posed in the classical sense, but also to important generalizations: the Cauchy problem for inclusion and the Cauchy problem for second order equations.Accessible to nonspecialists and beginning graduate students, this volume brings together many different ideas to serve as a reference on modern methods for abstract linear evolution equations.

Author Biography

Irina V. Melnikova is a Professor in the Department of Mathematics at Ural State University, Ekaterinburg, Russia Alexei Filinkov is a Professor in the Department of Pure Mathematics at Adelaide University in Australia

Table of Contents

Preface xi
Introduction xiii
Illustration and Motivation
1(16)
Heat equation
1(7)
The reversed Cauchy problem for the Heat equation
8(1)
Wave equation
9(8)
Semigroup Methods
17(104)
C0-semigroups
17(17)
Definitions and main properties
17(4)
The Cauchy problem
21(10)
Examples
31(3)
Integrated semigroups
34(20)
Exponentially bounded integrated semigroups
34(5)
(n,w)-well-posedness of the Cauchy problem
39(2)
Local integrated semigroups
41(9)
Examples
50(4)
κ-convoluted semigroups
54(6)
Generators of k-convoluted semigroups
54(5)
Θ-convoluted Cauchy problem
59(1)
C-regularized semigroups
60(20)
Generators of C-regularized semigroups
60(7)
C-well-posedness of the Cauchy problem
67(4)
Local C-regularized semigroups
71(3)
Integrated semigroups and C-regularized semigroups
74(3)
Examples
77(3)
Degenerate semigroups
80(11)
Generators of degenerate semigroups
80(2)
Degenrate 1-time integrated semigroups
82(4)
Maximal correctness class
86(3)
(n,w)-well-posedness of a degenerate Cauchy problem
89(1)
Examples
90(1)
The Cauchy problem for inclusions
91(18)
Multivalued linear operators
91(3)
Uniform well-posedness
94(5)
(n,w)-well-posedness
99(10)
Second order equations
109(12)
M;N-functions method
111(3)
Integrated semigroups method
114(3)
The degenerate Cauchy problem
117(4)
Abstract Distribution Methods
121(58)
The Cauchy problem
121(21)
Abstract (vector-valued) distributions
121(8)
Well-posedness in the space of distributions
129(9)
Well-posedness in the space of exponential distributions
138(4)
The degenerate Cauchy problem
142(19)
A-associated vectors and degenerate distribution semigroups
143(6)
Well-posedness in the sense of distributions
149(11)
Well-posedness in the space of exponential distributions
160(1)
Ultradistributions and new distributions
161(18)
Abstract ultradistributions
162(3)
The Cauchy problem in spaces of abstract ultradistributions
165(8)
The Cauchy problem in spaces of new distributions
173(6)
Regularization Methods
179(24)
The ill-posed Cauchy problem
179(11)
Quasi-reversibility method
179(6)
Auxiliary bounded conditions (ABC) method
185(3)
Carasso's method
188(2)
Regularization and regularized semigroups
190(13)
Comparison of the ABC and the quasi-reversibility methods
190(4)
`Differential' and variational methods of regularization
194(2)
Regularizing operators and local C-regularized semigroups
196(3)
Regularization of `slightly' ill-posed problems
199(4)
Bibliographic Remark 203(2)
Bibliography 205(24)
Glossary of Notation 229(4)
Index 233

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