Abstract Cauchy Problems: Three Approaches

by Filinkov; Alexei

• ISBN13: 9781584882503
• ISBN10: 1584882506
• Edition: 1st
• Format: Hardcover
• Copyright: 2001-03-27
• Publisher: Chapman & Hall/

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