The Linearization Method in Hydrodynamical Stability Theory

✍ Scribed by V. I. Yudovich

• Publisher: American Mathematical Society
• Year: 1989
• Tongue: English
• Leaves: 179
• Series: Translations of Mathematical Monographs, Vol. 74
• Edition: 0
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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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