Equations of Motion for Incompressible Viscous Fluids: With Mixed Boundary Conditions

Preface
Contents
1 Miscellanea of Analysis
	1.1 Banach Space, Fixed Point and Basics of Mapping
		1.1.1 Banach Space
		1.1.2 Fixed-Point Theorems
		1.1.3 Basics of Mappings
	1.2 Lebesgue Space, Convergence
		1.2.1 Lebesgue Space
		1.2.2 Convergence of Sequences of Functions
	1.3 Sobolev Space
		1.3.1 Definition of Sobolev Space
		1.3.2 Density and Continuation
		1.3.3 Imbedding
		1.3.4 Trace
		1.3.5 Some Inequalities
	1.4 Space of Abstract Functions
		1.4.1 Abstract Functions and Its Derivatives
		1.4.2 Compactness
	1.5 Operator Equations and Operator-Differential Equations
		1.5.1 Monotone Operator Equation
		1.5.2 Pseudo-Monotone Operator Equation
		1.5.3 Operator-Differential Equations
	1.6 Convex Functional
	1.7 Some Elementary Inequalities
	References
2 Fluid Equations
	2.1 Derivation of Equations for Fluid Motion
		2.1.1 Navier-Stokes Equations
		2.1.2 Equations of Motion for Fluid Under Consideration of Heat
	2.2 Boundary Conditions for the Navier-Stokes Equations
		2.2.1 Boundary Conditions on the Walls
		2.2.2 Boundary Conditions on Symmetric Planes
		2.2.3 Boundary Conditions on Inlets and Outlets
		2.2.4 Outflow Boundary Conditions on Imaginary Boundary
		2.2.5 Boundary Conditions on Free Surfaces
	2.3 Bilinear Forms for Hydrodynamics
		2.3.1 Bilinear Forms
		2.3.2 Variational Formulations for Mixed Boundary Value Problems of the Navier-Stokes Equations
	2.4 Bibliographical Remarks
		2.4.1 Fluid Equations
		2.4.2 Boundary Conditions of the Navier-Stokes Equations
		2.4.3 Bilinear Forms for Hydrodynamics
	References
3 The Steady Navier-Stokes System
	3.1 Properties on the Boundary Surfaces of Vector Fields
		3.1.1 The Second Fundamental Form and Shape Operator of Surface
		3.1.2 Properties on the Boundary Surface of Vector Fields
	3.2 Variational Formulations of the Steady Problems
	3.3 Existence of Solutions to the Steady Problems
	3.4 Bibliographical Remark
	References
4 The Non-steady Navier-Stokes System
	4.1 Existence of a Solution: The Case of Total Pressure
		4.1.1 Problem and Variational Formulation
		4.1.2 An Auxiliary Problem by Elliptic Regularization
		4.1.3 Proof of the Existence of a Solution
		4.1.4 The Stokes Problem
	4.2 Existence and Uniqueness of Solutions: The Case of Static Pressure
		4.2.1 Existence and Uniqueness of Solutions to Problem I
		4.2.2 Existence and Uniqueness of Solutions to Problem II
		4.2.3 Existence and Uniqueness of Solutions for Perturbed Data
	4.3 Bibliographical Remarks
	References
5 The Steady Navier-Stokes System with Friction Boundary Conditions
	5.1 Variational Formulations of Problems
		5.1.1 Variational Formulation: The Case of Static Pressure
		5.1.2 Variational Formulation: The Case of Total Pressure
		5.1.3 Variational Formulation: The Stokes Problem
	5.2 Solutions to Variational Inequalities
	5.3 Existence and Uniqueness of Solutions to the Steady Navier-Stokes Problems
	5.4 Bibliographical Remarks
	References
6 The Non-steady Navier-Stokes System with Friction Boundary Conditions
	6.1 Variational Formulations of Problems
		6.1.1 Variational Formulation: The Case of Total Pressure
		6.1.2 Variational Formulation: The Case of Static Pressure
		6.1.3 Variational Formulation: The Stokes Problem
	6.2 The Existence and Uniqueness of Solutions to Variational Inequalities
	6.3 Solutions to the Non-steady Navier-Stokes Problems
		6.3.1 Existence of a Solution: The Case of Total Pressure
		6.3.2 Existence of a Unique Solution: The Case of Static Pressure
		6.3.3 Existence of a Unique Solution: The Stokes Problem
	6.4 Bibliographical Remarks
	References
7 The Steady Boussinesq System
	7.1 Problems and Variational Formulations
		7.1.1 Variational Formulation: The Case of Static Pressure
		7.1.2 Variational Formulation: The Case of Total Pressure
	7.2 Existence and Uniqueness of Solutions: The Case of Static Pressure
		7.2.1 Existence of a Solution to an Auxiliary Problem
		7.2.2 Existence and Estimates of Solutions to the Approximate Problem
		7.2.3 Existence and Uniqueness of a Solution
	7.3 Existence of a Solution: The Case of Total Pressure
	7.4 Bibliographical Remarks
	References
8 The Non-steady Boussinesq System
	8.1 Problems and Assumptions
	8.2 Variational Formulations for Problems
		8.2.1 Variational Formulations: The Case of Static Pressure
		8.2.2 Variational Formulations: The Case of Total Pressure
	8.3 Existence and Uniqueness of Solutions: The Case of Static Pressure
		8.3.1 Existence and Estimation of Solutions to an Approximate Problem
		8.3.2 Existence and Uniqueness of a Solution
	8.4 Existence of a Solution: The Case of Total Pressure
		8.4.1 Existence of a Solution to an Approximate Problem
		8.4.2 Existence of a Solution
	8.5 Bibliographical Remarks
	References
9 The Steady Equations for Heat-Conducting Fluids
	9.1 Problems and Assumptions
	9.2 Variational Formulations for Problems
		9.2.1 Variational Formulation: The Case of Static Pressure
		9.2.2 Variational Formulation: The Case of Total Pressure
	9.3 Existence and Uniqueness of Solutions: The Case of Static Pressure
		9.3.1 Existence of a Solution to an Auxiliary Problem
		9.3.2 A Priori Estimates of Solutions to the Auxiliary Problem
		9.3.3 Passing to Limits
	9.4 Existence of a Solution: The Case of Total Pressure
	9.5 Bibliographical Remarks
	References
10 The Non-steady Equations for Heat-Conducting Fluids
	10.1 Problem and Variational Formulation
		10.1.1 Problem and Assumption
		10.1.2 Variational Formulation for Problem
	10.2 Existence of a Solution
		10.2.1 Existence of a Solution to an Approximate Problem
		10.2.2 Estimates of Solutions to the Approximate Problem
		10.2.3 Passing to the Limit
	10.3 Bibliographical Remarks
	References
Appendix  Index
Index