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