Stability, Instability, and Direct Integrals: 1st Edition (Paperback) book cover

Stability, Instability, and Direct Integrals

1st Edition

By B Scarpellini

Chapman and Hall/CRC

354 pages

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Paperback: 9780849306853
pub: 1999-01-29
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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.


"…introduction to…reaction-diffusion systems…suitable for a newcomer to the field."

Mathematical Reviews , 2000j

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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Chapman & Hall/CRC Research Notes in Mathematics Series

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Functional Analysis
SCIENCE / Mathematical Physics
SCIENCE / Physics