Stability, Instability, and Direct Integrals: (Chapman & Hall/CRC Research Notes in Mathematics Series)

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:
Preface Notations and Preliminaries Introduction 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 Comments 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 Remarks 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 Bibliography Index

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