This book presents the theory of the linearization method as applied to the problem of steady-state and periodic motions of continuous media. The author proves infinite-dimensional analogues of Lyapunov's theorems on stability, instability, and conditional stability for a large class of continuous media. In addition, semigroup properties for the linearized Navier-Stokes equations in the case of an incompressible fluid are studied, and coercivity inequalities and completeness of a system of small oscillations are proved.
Author(s): V. I. Yudovich
Series: Translations of Mathematical Monographs, Vol. 74
Edition: 0
Publisher: American Mathematical Society
Year: 1989
Language: English
Pages: C+vi+170+B
Cover
S Title
The Linearization Method in Hydrodynamical Stability Theory
Copyright ©1989 American Mathematical Society
ISBN 0-8218-4528-4
QA911.19313 1989 532'.5-dcl9
LCCN 89-315 CIP
Contents
Introduction
CHAPTER I Estimates of Solutions of the Linearized Navier-Stokes Equations
§1. Estimates of integral operators in Lp
1. Interpolation theorems
2. An extrapolation theorem and singular integrals.
3. Multipliers of series and Fourier integrals
§2. Some estimates of solutions of evolution equations
1. The Cauchy problem
2. Bounded, periodic, and almost periodic solutions
§3. Estimates of the "leading derivatives" of solutions of evolution equations
1. Estimates in Lp.
2. Some equations in Hilbert space
§4. Applications to parabolic equations and imbedding theorems
§5. The linearized Navier-Stokes equations
Appendix to §5
§6. An estimate of the resolvent of the linearized Navier-Stokes operator
§7. Estimates of the leading derivatives of a solution of the linearized steady-state Navier-Stokes equations
CHAPTER II Stability of Fluid Motion
§1. Stability of the motion of infinite-dimensional systems
§2. Conditions for stability
§3. Conditions for instability. Conditional stability
CHAPTER III Stability of Periodic Motions
§1. Formulation of the problem
§2. The problem with initial data
§3. A condition for asymptotic stability
§4. A condition for instability
§5. Conditional stability
§6. Stability of self-oscillatory regimes
§7. Instability of cycles
§8. Damping of the leading derivatives
Bibliography
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