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

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Equations of Motion for Incompressible Viscous Fluids: With Mixed Boundary ConditionsPalielināt
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This monograph explores the motion of incompressible fluids by presenting and incorporating various boundary conditions possible for real phenomena. The authors’ approach carefully walks readers through the development of fluid equations at the cutting edge of research, and the applications of a variety of boundary conditions to real-world problems. Special attention is paid to the equivalence between partial differential equations with a mixture of various boundary conditions and their corresponding variational problems, especially variational inequalities with one unknown. A self-contained approach is maintained throughout by first covering introductory topics, and then moving on to mixtures of boundary conditions, a thorough outline of the Navier-Stokes equations, an analysis of both the steady and non-steady Boussinesq system, and more. Equations of Motion for Incompressible Viscous Fluids is ideal for postgraduate students and researchers in the fields of fluid equations, numerical analysis, and mathematical modelling.

1 Miscellanea of Analysis
1(40)
1.1 Banach Space, Fixed Point and Basics of Mapping
1(4)
1.1.1 Banach Space
1(2)
1.1.2 Fixed-Point Theorems
3(1)
1.1.3 Basics of Mappings
4(1)
1.2 Lebesgue Space, Convergence
5(6)
1.2.1 Lebesgue Space
5(3)
1.2.2 Convergence of Sequences of Functions
8(3)
1.3 Sobolev Space
11(9)
1.3.1 Definition of Sobolev Space
12(2)
1.3.2 Density and Continuation
14(2)
1.3.3 Imbedding
16(2)
1.3.4 Trace
18(1)
1.3.5 Some Inequalities
19(1)
1.4 Space of Abstract Functions
20(8)
1.4.1 Abstract Functions and Its Derivatives
20(7)
1.4.2 Compactness
27(1)
1.5 Operator Equations and Operator-Differential Equations
28(5)
1.5.1 Monotone Operator Equation
28(2)
1.5.2 Pseudo-Monotone Operator Equation
30(1)
1.5.3 Operator-Differential Equations
31(2)
1.6 Convex Functional
33(4)
1.7 Some Elementary Inequalities
37(4)
References
38(3)
2 Fluid Equations
41(42)
2.1 Derivation of Equations for Fluid Motion
41(14)
2.1.1 Navier-Stokes Equations
41(8)
2.1.2 Equations of Motion for Fluid Under Consideration of Heat
49(6)
2.2 Boundary Conditions for the Navier-Stokes Equations
55(11)
2.2.1 Boundary Conditions on the Walls
58(3)
2.2.2 Boundary Conditions on Symmetric Planes
61(1)
2.2.3 Boundary Conditions on Inlets and Outlets
62(1)
2.2.4 Outflow Boundary Conditions on Imaginary Boundary
63(1)
2.2.5 Boundary Conditions on Free Surfaces
64(2)
2.3 Bilinear Forms for Hydrodynamics
66(10)
2.3.1 Bilinear Forms
67(4)
2.3.2 Variational Formulations for Mixed Boundary Value Problems of the Navier-Stokes Equations
71(5)
2.4 Bibliographical Remarks
76(7)
2.4.1 Fluid Equations
76(1)
2.4.2 Boundary Conditions of the Navier-Stokes Equations
76(1)
2.4.3 Bilinear Forms for Hydrodynamics
77(1)
References
77(6)
3 The Steady Navier-Stokes System
83(26)
3.1 Properties on the Boundary Surfaces of Vector Fields
83(11)
3.1.1 The Second Fundamental Form and Shape Operator of Surface
84(6)
3.1.2 Properties on the Boundary Surface of Vector Fields
90(4)
3.2 Variational Formulations of the Steady Problems
94(6)
3.3 Existence of Solutions to the Steady Problems
100(7)
3.4 Bibliographical Remark
107(2)
References
108(1)
4 The Non-steady Navier-Stokes System
109(40)
4.1 Existence of a Solution: The Case of Total Pressure
109(17)
4.1.1 Problem and Variational Formulation
110(3)
4.1.2 An Auxiliary Problem by Elliptic Regularization
113(5)
4.1.3 Proof of the Existence of a Solution
118(5)
4.1.4 The Stokes Problem
123(3)
4.2 Existence and Uniqueness of Solutions: The Case of Static Pressure
126(21)
4.2.1 Existence and Uniqueness of Solutions to Problem 1
127(11)
4.2.2 Existence and Uniqueness of Solutions to Problem II
138(3)
4.2.3 Existence and Uniqueness of Solutions for Perturbed Data
141(6)
4.3 Bibliographical Remarks
147(2)
References
148(1)
5 The Steady Navier-Stokes System with Friction Boundary Conditions
149(36)
5.1 Variational Formulations of Problems
150(17)
5.1.1 Variational Formulation: The Case of Static Pressure
152(12)
5.1.2 Variational Formulation: The Case of Total Pressure
164(2)
5.1.3 Variational Formulation: The Stokes Problem
166(1)
5.2 Solutions to Variational Inequalities
167(6)
5.3 Existence and Uniqueness of Solutions to the Steady Navier-Stokes Problems
173(9)
5.4 Bibliographical Remarks
182(3)
References
183(2)
6 The Non-steady Navier-Stokes System with Friction Boundary Conditions
185(42)
6.1 Variational Formulations of Problems
185(9)
6.1.1 Variational Formulation: The Case of Total Pressure
187(4)
6.1.2 Variational Formulation: The Case of Static Pressure
191(2)
6.1.3 Variational Formulation: The Stokes Problem
193(1)
6.2 The Existence and Uniqueness of Solutions to Variational Inequalities
194(21)
6.3 Solutions to the Non-steady Navier-Stokes Problems
215(9)
6.3.1 Existence of a Solution: The Case of Total Pressure
215(4)
6.3.2 Existence of a Unique Solution: The Case of Static Pressure
219(2)
6.3.3 Existence of a Unique Solution: The Stokes Problem
221(3)
6.4 Bibliographical Remarks
224(3)
References
225(2)
7 The Steady Boussinesq System
227(24)
7.1 Problems and Variational Formulations
227(8)
7.1.1 Variational Formulation: The Case of Static Pressure
231(2)
7.1.2 Variational Formulation: The Case of Total Pressure
233(2)
7.2 Existence and Uniqueness of Solutions: The Case of Static Pressure
235(12)
7.2.1 Existence of a Solution to an Auxiliary Problem
235(5)
7.2.2 Existence and Estimates of Solutions to the Approximate Problem
240(3)
7.2.3 Existence and Uniqueness of a Solution
243(4)
7.3 Existence of a Solution: The Case of Total Pressure
247(2)
7.4 Bibliographical Remarks
249(2)
References
250(1)
8 The Non-steady Boussinesq System
251(34)
8.1 Problems and Assumptions
251(2)
8.2 Variational Formulations for Problems
253(6)
8.2.1 Variational Formulations: The Case of Static Pressure
253(4)
8.2.2 Variational Formulations: The Case of Total Pressure
257(2)
8.3 Existence and Uniqueness of Solutions: The Case of Static Pressure
259(14)
8.3.1 Existence and Estimation of Solutions to an Approximate Problem
260(9)
8.3.2 Existence and Uniqueness of a Solution
269(4)
8.4 Existence of a Solution: The Case of Total Pressure
273(9)
8.4.1 Existence of a Solution to an Approximate Problem
273(5)
8.4.2 Existence of a Solution
278(4)
8.5 Bibliographical Remarks
282(3)
References
283(2)
9 The Steady Equations for Heat-Conducting Fluids
285(36)
9.1 Problems and Assumptions
285(1)
9.2 Variational Formulations for Problems
286(5)
9.2.1 Variational Formulation: The Case of Static Pressure
287(2)
9.2.2 Variational Formulation: The Case of Total Pressure
289(2)
9.3 Existence and Uniqueness of Solutions: The Case of Static Pressure
291(23)
9.3.1 Existence of a Solution to an Auxiliary Problem
291(7)
9.3.2 A Priori Estimates of Solutions to the Auxiliary Problem
298(11)
9.3.3 Passing to Limits
309(5)
9.4 Existence of a Solution: The Case of Total Pressure
314(4)
9.5 Bibliographical Remarks
318(3)
References
319(2)
10 The Non-steady Equations for Heat-Conducting Fluids
321(40)
10.1 Problem and Variational Formulation
321(6)
10.1.1 Problem and Assumption
321(2)
10.1.2 Variational Formulation for Problem
323(4)
10.2 Existence of a Solution
327(31)
10.2.1 Existence of a Solution to an Approximate Problem
327(14)
10.2.2 Estimates of Solutions to the Approximate Problem
341(10)
10.2.3 Passing to the Limit
351(7)
10.3 Bibliographical Remarks
358(3)
References
360(1)
Index 361
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