Operator Approach to Linear Problems of Hydrodynamics

The main topics presented in this book deal with methods from functional analysis applied to the study ofsmall movements and normal oscillations ofhydrome­ chanical systems having cavities filled with either ideal or viscous fluids. The book is a sequel to and at the same time substantially extends the volume entitled "Opera­ tor Methods in Linear Hydrodynamics: Evolution and Spectral Problems," by N. D. Kopachevsky, S.G. Krein, and Ngo Zuy Kan that was published in 1989 by the Nauka publishing house in Moscow. The present book includesseveral new problems on the oscillations ofpartially dissipative hydrosystems and the oscillations of visco-elastic or relaxing fluids. The contents of this book do not overlap almost at all with the ones in the following volumes: "Mathematical Problems of the Motion of Viscous Incopressible Fluids," by O. A. Ladyzhenskaya, "The Dynamics ofBodies with Cavities Filled with Fluids," by N. N. Moiseev and V. V. Rumiantzev, "Navier-Stokes Equations," by R. Temam, and "Boundary Problems for Navier-Stokes Equations," by S. M. Belonosov and K. A. Chernous. Mainly, the contents of the present book rely on the authors' and their students' works. We would like to express our gratitude to I. T. Gohberg and A. S. Markus, who encouraged us to publish the book and who offered many helpful suggestions. Our gratidude goes also to our colleagues T. Ya. Azizov, O. A. Ladyzhenskaya, N. N.

I: Mathematical Foundations of Linear Hydrodynamics.- 1: Operators on Hilbert Spaces.- 1.1 General Facts.- 1.1.1 The Concept of a Hilbert Space.- 1.1.2 The Space L2(?).- 1.1.3 Orthogonality. Projection onto a Subspace.- 1.1.4 Equivalent Norms.- 1.1.5 Linear Functionals. Riesz Theorem.- 1.1.6 Embeddings of Spaces. Riesz Theorem for Equipments.- 1.1.7 Orthonormal Systems and Bases.- 1.1.8 Bounded Linear Operators.- 1.1.9 Adjoint Operators.- 1.1.10 Self-Adjoint Operators.- 1.1.11 Self-Adjoint Compact Operators.- 1.1.12 Compact operators. s-numbers.- 1.1.13 Riesz Bases and p-Bases.- 1.1.14 Direct Sum of Subspaces. Invariant Subspaces.- 1.1.15 Eigen-and Associated (Root) Elements. Root Subspaces.- 1.1.16 Unbounded Linear Operators.- 1.1.17 Resolvent and Spectrum of a Linear Operator.- 1.1.18 Classification of Points in the Spectrum of a Linear Operator.- 1.1.19 Spectrum of a Self-Adjoint Operator. Weyl Theorem.- 1.1.20 Riesz Projections.- 1.1.21 Symmetric and Self-Adjoint Operators.- 1.1.22 Spectral Decomposition of Self-Adjoint Operators. Functions of Operators.- 1.1.23 Spaces with Degenerate Scalar Products. Seminorms.- 1.1.24 Equivalent Corrections of Seminorms.- 1.2 Sobolev Spaces.- 1.2.1 Finite Functions.- 1.2.2 Generalized Derivatives.- 1.2.3 The Definition of Sobolev Spaces.- 1.2.4 The Space L 1(?). Regions of the First Type.- 1.2.5 The Subspace Hó (?).- 1.2.6 Embedding H1(?) into L2(?).Regions of the Second Type.- 1.2.7 The Trace Operator. Regions of the Third Type.- 1.3 Spaces With Indefinite Metrics.- 1.3.1 J-Spaces.- 1.3.2 Uniformly Definite Subspaces.- 1.3.3 J-Orthonormal Systems and Bases.- 1.3.4 Linear Operators on J-Spaces.- 1.3.5 Invariant Subspaces of J-Self-Adjoint Operators.- 1.3.6 Pontryagin Spaces.- 1.3.7 On Completeness and Basicity for the System of Root Elements of a J-Self-Adjoint Operator.- 1.4 Eigenvalue Problems.- 1.4.1 Operator B is the Identity Operator.- 1.4.2 Operator B is Positive Definite.- 1.4.3 Positivity Condition for a Matrix Operator.- 1.4.4 Simplifying Equations with an Alternating Operator.- 1.4.5 Equations in Spaces with Indefinite Metrics.- 1.5 Evolution Equations in Hilbert Spaces.- 1.5.1 First Order Linear Differential Equations with Bounded Operator Coefficient.- 1.5.2 The Cauchy Problem for Equations with Unbounded Operators.- 1.5.3 Equations With a Negative Self-Adjoint Operator.- 1.5.4 Equations With a Dissipative Operator.- 1.5.5 Equations With Perturbed Operators.- 1.5.6 Stability.- 1.5.7 Nonhomogeneous Equations.- 1.5.8 Linear Differential Equations of the Second Order.- 1.5.9 Volterra Integral Equations.- 1.6 Spectral Theory of Operator Pencils.- 1.6.1 Eigen-and Associated Elements of an Operator Pencil.- 1.6.2 Root Functions.- 1.6.3 Fredholm Holomorphic Operator-Valued Functions.- 1.6.4 Linear Pencils. Theorems on Completeness of the System of Eigen-and Associated Elements.- 1.6.5 Keldysh-Type n-Multiple Completeness.- 1.6.6 Spectral Factorization of an Operator Pencil.- 1.6.7 Completeness With Finite Defect of a System of Eigen-and Associated Elements of an Operator-Valued Function.- 1.6.8 Asymptotic Behavior of Branches of Eigenvalues.- 1.6.9 Self-Adjoint Operator Pencils.- 1.6.10 On Riesz Basicity of the System of Eigenelements of a Self-Adjoint Operator-Valued Function.- 1.6.11 Variational Methods for Investigating Continuous Operator-Valued Functions.- 1.7 Asymptotic Methods for Solving Evolution Equations With a Small Parameter Attached to the Derivative.- 1.7.1 Equations With a Small Parameter Attached to the Derivative.- 1.7.2 Splitting of a Homogeneous Equation.- 1.7.3 Solvability of Commutator Equations.- 1.7.4 Asymptotic Expansions of Solutions.- 1.7.5 The Special Case of a Splitable Operator Kernel.- 1.7.6 The Case of a Nonstationary Perturbation.- 1.7.7 Nonhomogeneous Equations.- 1.7.8 Eigenvalue Problems.- 1.8 A General Scheme for Solving Boundary Value Problems.- 1.8.1 Hilbert Pairs. Generating Operators.- 1.8.2 Hilbert Pairs Connected With the Spaces H1(?) and L2(?).- 1.8.3 Hilbert Scale of Spaces. Space E-112.- 1.8.4 Self-Adjoint Extensions of Positive Definite Symmetric Operators. Generalized and Weak Solutions of Equations.- 1.8.5 Nonhomogeneous Boundary Value Problems.- 1.8.6 Spaces of Harmonic Functions.- 1.8.7 Embedding and Mapping. The Abstract Green Formula.- 2: Fundamental Spaces and Operators of Linear Hydrodynamics.- 2.1 Fundamental Spaces and Hydrodynamics Operators for an Ideal Fluid.- 2.1.1 Fields with Finite Kinetic Energy.- 2.1.2 Potential Fields.- 2.1.3 Divergence of Fields with Finite Kinetic Energy.- 2.1.4 The Space of Solenoidal Fields.- 2.1.5 Laplace Operator on the Space H1(?).- 2.1.6 Normal Component of a Field on the Boundary.- 2.1.7 Green Formula for the Laplace Operators. Harmonic Fields.- 2.1.8 Weyl Decomposition.- 2.1.9 Approximation by Smooth Fields.- 2.1.10 The Space of Velocity Fields for an Ideal Fluid in an Open Container.- 2.1.11 Systems of Nonmixing Fluids.- 2.1.12 Spaces of Velocity Fields for Systems of Nonmixing Ideal Fluids.- 2.2 Spaces and Hydrodynamics Operators for a Viscous Fluid...- 2.2.1 Forces of Internal Friction. Energy Dissipation.- 2.2.2 Divergence Operator. Solenoidal Fields.- 2.2.3 Vector Laplace Operator. Green Formula.- 2.2.4 Movement of a Viscous Fluid in a Closed Container. Korn Identity and Korn Inequality.- 2.2.5 Stokes Operator.- 2.2.6 Spaces of Velocity Fields for a Viscous Incompressible Fluid in an Open Container.- 2.2.7 Main Boundary Value Problems for the Fluid Movement in an Open Container.- 2.2.8 Spaces of Velocity Fields for a System of Viscous Fluids.- Appendix A: Remarks and Reference Comments to Part.- A.1 Chapter 1.- A.2 Chapter 2.- II: Motion of Bodies With Cavities Containing Ideal Fluids.- 3: Oscillations of a Heavy Ideal Fluid in Stationary and Nonstationary Containers.- 3.1 Equations of the Motion of a Rigid Body with a Cavity Filled with an Incompressible Fluid.- 3.1.1 Basic Concepts of Kinematics.- 3.1.2 Equations of Motion for an Incompressible Fluid.- 3.1.3 Boundary Conditions.- 3.1.4 Motion Equations for a Gyrostate.- 3.1.5 Dynamics Equations of the System “Body + Fluid” With a Partially Filled Cavity.- 3.1.6 Transition to Undimensional Variables.- 3.2 Motion of an Ideal Fluid in a Closed Stationary Container.- 3.2.1 Basic Equations.- 3.2.2 Existence of Solutions.- 3.3 Small movements of an Ideal Fluid in an Open Immovable Container.- 3.3.1 Statement of the Problem and the Basic Equations.- 3.3.2 Projection of Euler Equations.- 3.3.3 Existence of Solutions.- 3.3.4 Proper Oscillations.- 3.4 Small Joint Movements of a Fluid and a Container.- 3.4.1 Statement of the Problem and the Basic Equations.- 3.4.2 Finding the Velocity Field and the Pressure.- 3.4.3 Defining the Motion Law of the Body.- 3.4.4 Zhukovsky Potentials.- 3.5 Small Joint Movements Around a Fixed Point of a Body and a Fluid Partially Filling the Cavity.- 3.5.1 Statement of the Problem and the Basic Equations.- 3.5.2 The Law of Full Energy Balance.- 3.5.3 Projecting Euler Equations.- 3.5.4 Kinetic Moment Equation.- 3.5.5 Investigating the Complete System of Motion Equations.- 3.5.6 Proper Oscillations.- 3.5.7 Solving the Evolution Problem.- 3.6 Oscillations of a System of Fluids in an Immovable Container.- 3.6.1 Statement of the Problem.- 3.6.2 Orthogonal Projection Method.- 3.6.3 Transition to Operator Equation.- 3.6.4 Proper Osccilations.- 3.6.5 Small Movements of Stable Systems.- 4: Problems on Oscillations of Capillary Fluids and Problems on Hydroelasticity in Immovable Containers.- 4.1 Oscillations of a Capillary Fluid in a Rigid Container.- 4.1.1 On the Equilibrium State.- 4.1.2 Statement of the Problem on Small Oscillations.- 4.1.3 Law of Energy Balance.- 4.1.4 Transition to Operator Equation.- 4.1.5 Properties of the Potential Energy Operator.- 4.1.6 Proper Oscillations.- 4.1.7 Instability Conditions of the System.- 4.1.8 Solvability of the Evolution Problem.- 4.1.9 Oscillations of a System of Capillary Fluids.- 4.2 Oscillations of Fluids in Containers With Elastic Ends.- 4.2.1 Solving the Static Problem.- 4.2.2 Formulation of the Problem on Small Oscillations.- 4.2.3 Energy-Preserving Law.- 4.2.4 Operator Equation of the Problem.- 4.2.5 Properties of the Operators of the Problem.- 4.2.6 Proper Oscillations.- 4.2.7 Evolution Problem.- 4.2.8 Oscillations of a Fluid in a Container With One Elastic End.- 4.2.9 Oscillations of a System of Fluids in a Container With Elastic Plane.- 4.3 Oscillations of a Fluid in a Partially Filled Container With an Elastic Bottom.- 4.3.1 Determining the Equilibrium State.- 4.3.2 Formulation of the Problem on Small Oscillations.- 4.3.3 On Solvability of the Evolution and Spectral Problems.- 4.3.4 Oscillations of a Capillary Fluid in a Partially Filled Elastic Container.- 4.3.5 Systems of Heavy Fluids in a Container With Elastic Plates.- 4.3.6 Systems of Capillary Fluids in a Container With Elastic Ends.- 4.3.7 Compound Systems of Plate-Partitions and Nonmixing Fluids.- 5: Other Operator Approaches to Hydrodynamics Problems of Ideal Fluids.- 5.1 Plane Problems on Proper Oscillations of a Heavy Fluid in a Channel. An Application of the Stream Function.- 5.1.1. Spectral Problem for the Stream Function.- 5.1.2. Properties of Nodal Lines of Stream Eigenfunctions….- 5.1.3. Estimates of Eigenvalues.- 5.2 Shallow Water Theory in Problems on Oscillations of Heavy Ideal Fluids in Bounded Regions.- 5.2.1. Formulation of the Problem with a Small Parameter.- 5.2.2. Asymptotic Solution in First Approximation.- 5.2.3. Formulas for Calculating Second Order Approximations.- 5.2.4. Plane Problems.- 5.2.5. Examples.- 5.2.6. Systems of Nonmixing Fluids.- 5.3 Oscillations of a System “Fluid Gas” in a Bounded Region.- 5.3.1. Formulation of the Initial Boundary Value Problem.- 5.3.2. Formulation of the Spectral Problem.- 5.3.3. Transition to a System of Operator Equations.- 5.3.4 Theorem on Spectrum.- 5.3.5 Variational Principles for Eigenvalues.- 6: Oscillations of an Ideal Rotating Fluid.- 6.1 Motion of Fluids in Rotating Containers.- 6.1.1 Statement of the Problem and the Main Equations.- 6.1.2. Existence of Solutions.- 6.1.3 Normal Oscillations.- 6.2 Motion of a Gyrostate Similar to Uniform Rotation About a Fixed Axis.- 6.2.1 Statement of the Problem.- 6.2.2 Transition to the Evolution Equation in a Hilbert Space.- 6.2.3 Properties of the Operators in the Problem.- 6.2.4 Existence of Solution to the Boundary Value Problem...- 6.3 Rotation of a Fluid in a Partially Filled Container.- 6.3.1 On the Equilibrium State.- 6.3.2 Statement of the Problem on Small Oscillations.- 6.3.3 Method of Orthogonal Projection.- 6.3.4 Properties of the Operators of the Problem.- 6.3.5 Systems of Nonmixing Fluids.- 6.3.6 Transition to an Operator Equation and Properties of the Operators of the Problem.- 6.4 Solving the Initial Boundary Value Problem.- 6.4.1 Generalized Solution of the Operator Equation.- 6.4.2 Small Movements of Fluid in a Partially Filled Container.- 6.4.3 On the Structure of the Spectrum of a Vortical Operator.- 6.4.4 Classes of Free Movements.- 6.4.5 Free Movements of a System of Fluids.- 6.5 Self-adjoint Operator Pencils Generated by Problems on Oscillations of a Rotating Ideal Fluid.- 6.5.1 The Main Operator Pencil.- 6.5.2 On the Spectrum of the Operator Pencil.- 6.5.3 Operator Pencils with Analytic Perturbations.- 6.5.4 Factorization of the Operator Pencil.- 6.5.5 Systems of Eigenelements With Defect Basicity.- 6.5.6 Double-Sided Estimates of Positive and Negative Eigenvalues.- 6.5.7 On the Essential Spectrum of the Problem.- 6.6 Proper Oscillations of a Rotating Fluid.- 6.6.1 Surface and Internal Waves.- 6.6.2 Properties of the Surface Waves.- 6.6.3 On Existence and Properties of Internal Waves.- 6.6.4 Oscillations of a System of Nonmixing Fluids.- Appendix B: Remarks and Reference Comments to Part II.- B.1 Chapter 3.- B.2 Chapter 4.- B.3 Chapter 5.- B.4 Chapter 6.- Standard Reference Texts.- Standard Reference Texts.- List of Symbols.