Operator Approach to Linear Problems of Hydrodynamics

As mentioned in the Introduction to Volume I, the present monograph is intended both for mathematicians interested in applications of the theory of linear operators and operator-functions to problems of hydrodynamics, and for researchers of applied hydrodynamic problems, who want to study these problems by means of the most recent achievements in operator theory. The second volume considers nonself-adjoint problems describing motions and normal oscillations of a homogeneous viscous incompressible fluid. These ini­ tial boundary value problems of mathematical physics include, as a rule, derivatives in time of the unknown functions not only in the equation, but in the boundary conditions, too. Therefore, the spectral problems corresponding to such boundary value problems include the spectral parameter in the equation and in the bound­ ary conditions, and are nonself-adjoint. In their study, we widely used the theory of nonself-adjoint operators acting in a Hilbert space and also the theory of operator pencils. In particular, the methods of operator pencil factorization and methods of operator theory in a space with indefinite metric find here a wide application. We note also that this volume presents both the now classical problems on oscillations of a homogeneous viscous fluid in an open container (in an ordinary state and in weightlessness) and a new set of problems on oscillations of partially dissipative hydrodynamic systems, and problems on oscillations of a visco-elastic or relaxing fluid. Some of these problems need a more careful additional investigation and are rather complicated.

Table of Contents Volume II.- III: Motion of Bodies with Cavities ContainingViscous Incompressible Fluids.- 7: Motion of Bodies with Cavities Completely Filled with Viscous Incompressible Fluids.- 7.1 Motion of Fluids Completely Filling a Cavity in aStationary Body.- 7.1.1 Statement of the Problem and the Basic Equations.- 7.1.2 Reducing the Problem to a Differential Equation in aHilbert Space. Existence of Solutions.- 7.1.3 Structure of the Spectrum of the Problem.- 7.1.4 Perturbation of the Stationary Motion of a Fluid.- 7.1.5 Small Movements of a Fluid in a Rotating Container.- 7.2 Small Movements of a Gyrostate Around a Fixed Mass Center.- 7.2.1 Statement of the Problem and the Basic Equations.- 7.2.2 Transition to a Differential Equation in a Hilbert Space.- 7.2.3 Properties of the Translation Operator.- 7.2.4 Existence of Solutions of the Evolution Problem.- 7.2.5 Normal Oscillations.- 7.3 Rotating Motion of a Gyrostate.- 7.3.1 Statement of the Problem and the Basic Equations.- 7.3.2 Transition to a Differential Equation in a Hilbert Space.- 7.3.3 Properties of the Operators of the Problem.- 7.3.4 Normal Oscillations.- 7.3.5 Solvability of the Nonstationary Problem.- 7.4 Asymptotic Solutions for High Viscosity.- 7.4.1 Solving the Hydrodynamics Problem.- 7.4.2 Asymptotic Equations of the Motion of a Rigid Body.- 7.4.3 An Example.- 7.5 Oscillations of a Pendulum With a Cavity Completely Filled With aViscous Fluid.- 7.5.1 Towards the Statement of the Problem.- 7.5.2 Transition to a System of Operator Equations.- 7.5.3 The Indefinite Metric Approach.- 7.5.4 Other Properties of Solutions of the Spectral Problem.- 7.5.5 On Riesz and p-Basicity of Modes of Dissipative Waves.- 7.6 Problems on Fluids Flowing Through a Given Cavity.- 7.6.1 The Basic Equations.- 7.6.2 Application of the Abstract Scheme.- 7.6.3 Transition to Operator Equations inOrthogonal Subspaces.- 7.6.4 Theorem on Existence of a Generalized Solution.- 7.7 Convective Movements of Fluids in a Closed Cavity.- 7.7.1 Equations of Thermal Convection.- 7.7.2 Conditions of Mechanical Equilibrium.- 7.7.3 Final Statement of the Problem.- 7.7.4 Transition to an Operator Equation.- 7.7.5 Solvability of the Initial Boundary Value Problem.- 7.7.6 Normal Movements of a System Heated from Below.- 7.7.7 Normal Oscillations for Heating from Above.- 7.7.8 On Transition of Eigenvalues to the Left Half-Plane in theCase of Heating from Below.- 8: Motion of Viscous Fluids in Open Containers.- 8.1 Small Movements of Viscous Fluids in an OpenImmovable Container.- 8.1.1 Classical Statement of the Problem.- 8.1.2 Auxiliary Boundary Value Problems.- 8.1.3 Generalized Solutions of the Homogeneous NonstationaryProblem.- 8.1.4 Motions with Small Mass Forces.- 8.1.5 Equation of Energy Balance.- 8.1.6 Equation of Normal Oscillations.- 8.2 The Main Operator Pencil.- 8.2.1 Structure of the Spectrum of the Problem.- 8.2.2 Linearization of the Pencil.- 8.2.3 Mutual Relationships between Eigen-andAssociated Elements of the Two Pencils.- 8.2.4 Transformation to a Nondegenerate Pencil.- 8.2.5 The Property of Two-Multiple Basicity.- 8.2.6 Spectral Factorization of the Overdamped Pencil.Separate Basicity.- 8.2.7 Double-Sided Inequalities for the Two Branches ofEigenvalues.- 8.2.8 The General Case.- 8.3 Normal Oscillations and the Spectrum of the HydrodynamicsProblem.- 8.3.1 General Properties of the Spectrum.- 8.3.2 Influence of Fluid Viscosity on the Structure of theSpectrum of the Problem.- 8.3.3 Properties of Surface and Internal Waves.- 8.3.4 On the Basicity of Modes of Normal Oscillations.- 8.4 Oscillations of a Heavy Rotating Fluid.- 8.4.1 Statement of the Problem.- 8.4.2 Transition to a System of Operator Equations.- 8.4.3 Solvability of the Nonstationary Problem.- 8.4.4 Equations of Normal Oscillations.- 8.4.5 Investigation of the Spectral Problem.- 8.4.6 On the Completeness of Systems of Modes of NormalOscillations of the Initial Problem.- 8.5 Asymptotic Solutions for High Viscosity.- 8.5.1 The Cauchy Problem.- 8.5.2 Normal Oscillations.- 8.5.3 Motions under Mass Forces.- 8.5.4 The Case of Rotating Viscous Fluids.- 8.6 Oscillations of a System of Nonmixing Fluids.- 8.6.1 Statement of the Problem on Small Oscillations.- 8.6.2 Transition to a System of Operator Equations.- 8.6.3 The Theorem on Existence of a Generalized Solution..- 8.6.4 Normal Oscillations of a System of Fluids.- 8.6.5 On the Stability of the Relative Equilibrium State.- 8.7 Small Motions Around a Fixed Point of a Body with aCavity Partially Filled with Fluid.- 8.7.1 Basic Equations.- 8.7.2 Transition to a System of Operator Equations.- 8.7.3 Auxiliary Results.- 8.7.4 Existence of Solution of the Boundary Value Problem.- 8.8 Normal Oscillations of a Pendulum Partially Filled with aFluid (The Plane Problem).- 8.8.1 Statement of the Problem.- 8.8.2 Transition to a Differential Equation in aHilbert Space.- 8.8.3 Properties of Operator Coefficients of the EvolutionEquation.- 8.8.4 Normal Oscillations. Properties of the SpectralProblem.- 8.8.5 Theorem on Instability.- 8.8.6 On the Solvability of the Initial Boundary Value Problem.- 8.9 Convection in a Partially Filled Container.- 8.9.1 Statement of the Problem.- 8.9.2 Transition to a System of Operator Equations.- 8.9.3 Solvability of the Evolution Problem.- 8.9.4 Normal Convective Movements. Reduction to anOperator Pencil.- 8.9.5 Dissipatively Thermal and Surface Waves under theGeneral Law of Heat Transfer.- 8.9.6 Surface and Internal Waves for Heating From Above.- 8.9.7 Normal Oscillations for Heating from Below and for aGiven Heat Flow on the Free Surface.- 8.10 Sufficient Conditions of Instability for ConvectiveMovements of a Fluid.- 8.10.1 Transition to a Two-Parameter Pencil.- 8.10.2 On the Structure of the Kernels of OperatorCoefficients.- 8.10.3 On the Existence of Eigenvalues in the Left ComplexHalf-Plane.- 8.10.4 Double-Sided Estimates for Eigenvalues.- 8.10.5 Derivation of a Sufficient Condition for Instability.- 8.10.6 Remarks.- 9: Oscillations of Capillary Viscous Fluids.- 9.1 Statement of the Problem.- 9.1.1 Basic Equations and Boundary Conditions.- 9.1.2 Some Properties of Solutions to the NormalOscillation Problem.- 9.1.3 On the Spectrum Structure of Normal Oscillations.- 9.2 Oscilations of Capillary Fluids in Arbitrary Containers.- 9.2.1 Transition to a System of Operator Equations.- 9.2.2 Normal Oscillations. Properties of the Operators of theOscillations Problem.- 9.2.3 Normal Oscillations of a Rotating Fluid.- 9.2.4 Normal Oscillations of a Nonrotating Fluid.- 9.2.5 The Matrix Structure of the Main Operator.- 9.2.6 On the Finiteness of the Number of Nonreal Eigenvalues.- 9.2.7 Heuristic Considerations. The AbstractSpectral Problem.- 9.2.8 Heuristic Considerations. Physical Conclusions.- 9.2.9 The Solvability of the Evolution Problem.- 9.3 The Inverse of the Lagrange Theorem on Stability.- 9.3.1 Formulating the Theorem.- 9.3.2 Auxiliary Propositions.- 9.3.3 The Principle of Changing Stability.- 9.3.4 Transition to an Equation with a Compact Operator.- 9.3.5 Application of Perturbation Theory.- 9.3.6 The Existence of Eigenvalues in the Left ComplexHalf-Plane for an Arbitrary Viscosity Value.- 9.4 Motions of a Rigid Body Containing a Cavity Filled with aCapillary Fluid under Conditions of Complete Low Gravity.- 9.4.1 Statement of the Problem.- 9.4.2 Transition to a System of Operator Equations.- 9.4.3 Normal Oscillations. Transition to an Equation with aDissipative Operator.- 9.4.4 Properties of Normal Movements.- 9.4.5 The Existence of a Generalized Solution to theNonstationary Problem.- Appendix C: Remarks and Reference Comments to Part.- C.1 Chapter 7.- C.2 Chapter 8.- C.3 Chapter 9.- IV: Small Oscillations of Complex Hydrodynamic Systems.- 10: Oscillations of Partially Dissipative Hydrosystems.- 10.1 Statement of the Problem.- 10.1.1 The Classical Statement of the Problem.- 10.1.2 The Law of Full Energy Balance. Definition of aGeneralized Solution.- 10.1.3 Normal Oscillations. Statement of the Problem.- 10.2 Studying an Initial Boundary Value Problem.- 10.2.1 Projections of Euler and Navier-Stokes Equationson Orthogonal Subspaces.- 10.2.2 Auxiliary Boundary Value Problems.- 10.2.3 Properties of Matrix Blocks and TheirPhysical Meanings.- 10.2.4 Theorem on Correct Solvability of the InitialBoundary Value Problem.- 10.3 Model Problem on Normal Oscilations of PartiallyDissipative Hydrosystems.- 10.3.1 Statement of the Model Problem.- 10.3.2 Obtaining the Characteristic Equation.- 10.3.3 Studying the Characteristic Equation.- 10.3.4 General Conclusions and Hypotheses on the Structureof the Spectrum of the Hydrodynamics Problem.- 10.4 Normal Oscillations of a Partially DissipativeHydrosystem in an Arbitrary Domain.- 10.4.1 Transition to an Operator Pencil with BoundedOperator Coefficients.- 10.4.2 General Properties of the Spectrum.- 10.4.3 The Theorem on Spectrum Location.- 10.5 On the Completeness of the System of Modes of NormalOscillations.- 10.5.1 Auxiliary Results.- 10.5.2 Theorem on Completeness. Keldysh SchemeRealization.- 10.5.3 On the Existence of Branches of Eigenvalues with aLimit Point at Infinity. Heuristic Arguments.- 10.5.4 Concluding Remarks.- 11: Oscillations of Visco-Elastic and Relaxing Media.- 11.1 Visco-Elastic Fluids in Completely Filled Containers.- 11.1.1 A Model of a Visco-Elastic Fluid.- 11.1.2 Statement of the Initial Boundary Value Problem. 318 11.1.3 On the Solvability of the Initial BoundaryValue Problem.- 11.1.4 Normal Oscillations.- 11.2 Abstract Evolution and Spectral Problems Generated bySmall Motions of a Visco-Elastic Fluid.- 11.2.1 Statement of the Problem. Transition to anEquation with a Dissipative Operator.- 11.2.2 On the Solvability of the Cauchy Problem forIntegro-Differential Equations.- 11.2.3 Spectral Problem. Transition to an Equation with aBounded Operator.- 11.2.4 Properties of Operator Coefficients of the SpectralProblem.- 11.2.5 Properties of Solutions of the Spectral Problem.- 11.3 Small Motions and Normal Oscillations of a Visco-ElasticFluid in an Open Container.- 11.3.1 Mathematical Statement of the Problem.- 11.3.2 Transition to a System of Operator Equations.- 11.3.3 On the Solvability of the Initial Boundary ValueProblem.- 11.3.4 Normal Oscillations. Main Operator Pencil.- 11.4 Multiple Basicity of the System of Eigen-and Associated Elements for the Problem on Normal Oscillations of aVisco-Elastic Fluid in an Open Container.- 11.4.1 The Linearization of the Pencil.- 11.4.2 The Theorem on Basicity of the System of Root Elementsof the Linear Pencil.- 11.4.3 Connection between the System of Root Elementsof the Two Problems.- 11.4.4 The Theorem on Basicity of Special Form Elements.- 11.5 Additional Properties of Solutions of the Spectral Problem..- 11.5.1 On the Existence of Different Branches of Eigenvalues.- 11.5.2 On the Location of Nonreal Eigenvalues in theComplex Plane.- 11.5.3 Multiple Basicity and p-Basicity. An IndefiniteMetric Approach Using Krein Space Theory.- 11.6 Oscillations of Relaxing Fluids.- 11.6.1 Classical Statement of the Problem on Small Motionsof a Relaxing Fluid.- 11.6.2 Transition to an Initial Boundary Value Problem forOne Scalar Function.- 11.6.3 The Simplest Problem on Oscilations of a RelaxingFluid.- 11.6.4 On the Solvability of the Cauchy Problem for anAbstract Integro-Differential Equation Connected withSmall Motions of a Relaxing Fluid.- 11.6.5 Normal Oscilations of a Relaxing Fluid withVariable Medium Characteristics.- 11.6.6 Physical Conclusions.- Appendix D: Remarks and Reference Comments to Part.- D.1 Chapter 10.- D.2 Chapter 11.- Standard Reference Texts.- List of Symbols.