Stability, Instability, and Direct Integrals  book cover
1st Edition

Stability, Instability, and Direct Integrals

ISBN 9780849306853
Published January 29, 1999 by Chapman and Hall/CRC
354 Pages

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Book Description

In this masterful study, the author sets forth a unique treatment of the stability and instability of the periodic equilibria of partial differential equations as they relate to the notion of direct integrals. His results, and to a large extent his methods are new. Although he aims this work at theory rather than applications, once the theoretical framework is built, applications emerge with ease.
Readers with some basis in functional analysis-notably semigroups-and measure theory can strengthen their background through its introductory material on direct integrals and its proofs worked out in detail. In Stability, Instability and Direct Integrals, applied and pure mathematicians and theoretical physicists can discover from an acknowledged innovator the most recent results of research in this active and expanding field.

Table of Contents

Notations and Preliminaries
Reaction-Diffusion Systems on an Infinite Plate
Reaction-Diffusion Systems on an Infinite Plate
The Laplacian on an infinite plate
Floquet-Periodic Functions
q-Periodic Functions
Fourier Series
The q-Periodic Laplacian
The Periodic Case
Direct Integrals
Direct Integrals of Hilbert Spaces
Direct Integrals of Operators
Spectral Considerations
Relations Between Measure and Spectra
Holomorphic Families of Operators
Proof of Lemma 4.2
Navier-Stokes on an Infinite Plate
Navier-Stokes on an Infinite Plate
The Stokes Operator on the Infinite Plate
Fourier Expansions
The Projection Operator P
The Floquet-Periodic Stokes Operator
Parity Considerations
Traces Expressed in Terms of Fourier Series
The q-Periodic Stokes Operator
Computational Aspects
Some Auxiliary Lemmas
The Regularity Proof
The Proof of Theorem 6.2 (a)
Discussion of the Regularity Proof
Some Consequences of the Regularity Proof
The q-Periodic Projection Operator
A Different Definition of Eq
The q-Periodic Neumann Problem
The q-Periodic Projection Operator
Stokes Operator, Pressure and Direct Integrals
Preparatory Steps
Proof of Theorem 8.1
Proof of Theorem 8.2
Parity Reconsidered
Spectral Theory and Direct Integrals
Some Holomorphic Families of Operators
Families of Resolvents
Local Spectral Relations
The Corners: Preliminary Remarks
The Corners and their Influence on the Spectrum
Relationship with the Periodic Spectrum
The Principle of Linearized Instability
Remarks on the Principle of Linearized Instability
Real Elements in the Space of Direct Integrals
A Topological Interpretation
Construction of a Family of Projection Operators
Direct Integral Representation of a Projection Operator
The Principle of Linearized Instability: Nonlinear Part
The Nonlinear Terms
Further Remarks on Fractional Powers
Fixed Points of an Integral Equation
Instability: Proof of Theorem 10.2**
Instability via Complex Projection Operators
Further Remarks
The Boussinesq Equations
The Boussinesq Equations
Remarks in the Infinite Plate
Remarks in the q-Periodic Setting
Remarks on Direct Integrals
Remarks on Spectral Theory
Remarks on the Principle of Linearized Instability

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"…introduction to…reaction-diffusion systems…suitable for a newcomer to the field."
Mathematical Reviews , 2000j