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The Linearization Method in Hydrodynamical Stability Theory
V. I. Yudovich

Translations of Mathematical Monographs
1989; 170 pp; softcover
Volume: 74
Reprint/Revision History:
reprinted 1991
ISBN-10: 0-8218-4528-4
ISBN-13: 978-0-8218-4528-8
List Price: US$75
Member Price: US$60
Order Code: MMONO/74
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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

  • Estimates of solutions of the linearized Navier-Stokes equations
  • Estimates of integral operators in \(L_p\)
  • Some estimates of solutions of evolution equations
  • Estimates of the "leading derivatives" of solutions of evolution equations
  • Applications to parabolic equations and imbedding theorems
  • The linearized Navier-Stokes equations
  • An estimate of the resolvent of the linearized Navier-Stokes operator
  • Estimates of the leading derivatives of a solution of the linearized steady-state Navier-Stokes equations
  • Stability of fluid motion
  • Stability of the motion of infinite-dimensional systems
  • Conditions for stability
  • Conditions for instability. Conditional stability
  • Stability of periodic motions
  • Formulation of the problem
  • The problem with initial data
  • A condition for asymptotic stability
  • A condition for instability
  • Conditional stability
  • Stability of auto-oscillatory regimes
  • Instability of cycles
  • Damping of the leading derivatives
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