Operator Approach to Linear Problems of Hydrodynamics

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 Ho (?).- 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.