Stability, instability, and direct integrals
Bruno Scarpellini · 1999
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