دانلود کتاب The Navier-Stokes Problem in the 21st Century

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Cover
Half Title
Title Page
Copyright Page
Contents
Preface to the First Edition
Preface to the Second Edition
1. Presentation of the Clay Millennium Prizes
1.1. Regularity of the Three-Dimensional Fluid Flows: A Mathematical Challenge for the 21st Century
1.2. The Clay Millennium Prizes
1.3. The Clay Millennium Prize for the Navier–Stokes Equations
1.4. Boundaries and the Navier–Stokes Clay Millennium Problem
2. The Physical Meaning of the Navier–Stokes Equations
2.1. Frames of References
2.2. The Convection Theorem
2.3. Conservation of Mass
2.4. Newton’s Second Law
2.5. Pressure
2.6. Strain
2.7. Stress
2.8. The Equations of Hydrodynamics
2.9. The Navier–Stokes Equations
2.10. Vorticity
2.11. Boundary Terms
2.12. Blow-up
2.13. Turbulence
3. History of the Equation
3.1. Mechanics in the Scientific Revolution Era
3.2. Bernoulli’s Hydrodymica
3.3. D’Alembert
3.4. Euler
3.5. Laplacian Physics
3.6. Navier, Cauchy, Poisson, Saint-Venant and Stokes
3.7. Reynolds
3.8. Oseen, Leray, Hopf and Ladyzhenskaya
3.9. Turbulence Models
4. Classical Solutions
4.1. The Heat Kernel
4.2. The Poisson Equation
4.3. The Helmholtz Decomposition
4.4. The Stokes Equation
4.5. The Oseen Tensor
4.6. Classical Solutions for the Navier–Stokes Problem
4.7. Maximal Classical Solutions and Estimates in L∞ Norms
4.8. Small Data
4.9. Spatial Asymptotics
4.10. Spatial Asymptotics for the Vorticity
4.11. Maximal Classical Solutions and Estimates in L2 Norms
4.12. Intermediate Conclusion
5. A Capacitary Approach of the Navier–Stokes Integral Equations
5.1. The Integral Navier–Stokes Problem
5.2. Quadratic Equations in Banach Spaces
5.3. A Capacitary Approach of Quadratic Integral Equations
5.4. Generalized Riesz Potentials on Spaces of Homogeneous Type
5.5. Dominating Functions for the Navier–Stokes Integral Equations
5.6. Oseen’s Theorem and Dominating Functions
5.7. Functional Spaces and Multipliers
6. The Differential and the Integral Navier–Stokes Equations
6.1. Very Weak Solutions for the Navier–Stokes Equations
6.2. Heat Equation
6.3. The Leray Projection Operator
6.4. Stokes Equations
6.5. Oseen Equations
6.6. Mild Solutions for the Navier–Stokes Equations
6.7. Suitable Solutions for the Navier–Stokes Equations
7. Mild Solutions in Lebesgue or Sobolev Spaces
7.1. Kato’s Mild Solutions
7.2. Local Solutions in the Hilbertian Setting
7.3. Global Solutions in the Hilbertian Setting
7.4. Sobolev Spaces
7.5. A Commutator Estimate
7.6. Lebesgue Spaces
7.7. Maximal Functions
7.8. Basic Lemmas on Real Interpolation Spaces
7.9. Uniqueness of L3 Solutions
8. Mild Solutions in Besov or Morrey Spaces
8.1. Morrey Spaces
8.2. Morrey Spaces and Maximal Functions
8.3. Uniqueness of Morrey Solutions
8.4. Besov Spaces
8.5. Regular Besov Spaces
8.6. Triebel–Lizorkin Spaces
8.7. Fourier Transform and Navier–Stokes Equations
8.8. The Cheap Navier–Stokes Equation
8.9. Plane Waves
9. The Space BMO−1 and the Koch and Tataru Theorem
9.1. The Koch and Tataru Theorem
9.2. A Variation on the Koch and Tataru Theorem
9.3. Q-spaces
9.4. A Special Subclass of BMO−1
9.5. Ill-posedness
9.6. Further Results on Ill-posedness
9.7. Large Data for Mild Solutions
9.8. Stability of Global Solutions
9.9. Analyticity
9.10. Small Data
10. Special Examples of Solutions
10.1. Symmetries for the Navier–Stokes Equations
10.2. Two-and-a-Half Dimensional Flows
10.3. Axisymmetrical Solutions
10.4. Helical Solutions
10.5. Brandolese’s Symmetrical Solutions
10.6. Self-similar Solutions
10.7. Stationary Solutions
10.8. Landau’s Solutions of the Navier–Stokes Equations
10.9. Time-Periodic Solutions
10.10. Beltrami Flows
11. Blow-up?
11.1. First Criteria
11.2. Blow-up for the Cheap Navier–Stokes Equation
11.3. Serrin’s Criterion
11.4. A Remark on Serrin’s Criterion and Leray’s Criterion
11.5. Some Further Generalizations of Serrin’s Criterion
11.6. Vorticity
11.7. Squirts
11.8. Eigenvalues of the Strain Matrix
12. Leray’s Weak Solutions
12.1. The Rellich Lemma
12.2. Leray’s Weak Solutions
12.3. Weak-Strong Uniqueness: The Prodi–Serrin Criterion
12.4. Weak-Strong Uniqueness and Morrey Spaces on the Product Space R × R3
12.5. Almost Strong Solutions
12.6. Weak Perturbations of Mild Solutions
12.7. Non-uniqueness of Weak Solutions
12.8. The Inviscid Limit
13. Partial Regularity Results for Weak Solutions
13.1. Interior Regularity
13.2. Serrin’s Theorem on Interior Regularity
13.3. O’Leary’s Theorem on Interior Regularity
13.4. Further Results on Parabolic Morrey Spaces
13.5. Hausdorff Measures
13.6. Singular Times
13.7. The Local Energy Inequality
13.8. The Caffarelli-Kohn-Nirenberg Theorem on Partial Regularity
13.9. Proof of the Caffarelli–Kohn–Nirenberg Criterion
13.10. Parabolic Hausdorff Dimension of the Set of Singular Points
13.11. On the Role of the Pressure in the Caffarelli, Kohn, and Nirenberg Regularity Theorem
14. A Theory of Uniformly Locally L2 Solutions
14.1. Uniformly Locally Square Integrable Solutions
14.2. Local Inequalities for Local Leray Solutions
14.3. The Caffarelli, Kohn and Nirenberg ϵ–Regularity Criterion
14.4. A Weak-Strong Uniqueness Result
14.5. Global Existence for Local Leray Solutions
14.6. Weighted Estimates
14.7. A Stability Estimate
14.8. Barker’s Theorem on Weak-Strong Uniqueness
14.9. Further Results on Global Existence of Suitable Weak Solutions
15. The L3 Theory of Suitable Solutions
15.1. Local Leray Solutions with an Initial Value in L3
15.2. Blow up in Finite Time
15.3. Backward Uniqueness for Local Leray Solutions
15.4. Seregin’s Theorem
15.5. Further Comments on Seregin’s Theorem
15.6. Critical Elements for the Blow-up of the Cauchy Problem in L3
15.7. Known Results on the Cauchy Problem for the Navier–Stokes Equations in Presence of a Force
15.8. Local Estimates for Suitable Solutions
15.9. Uniqueness for Suitable Solutions
15.10. A Quantitative One-scale Estimate for the Caffarelli–Kohn–Nirenberg Regularity Criterion
15.11. The Topological Structure of the Set of Suitable Solutions
15.12. Escauriaza, Seregin and Šverák’s Theorem
16. Self-similarity and the Leray–Schauder Principle
16.1. The Leray–Schauder Principle
16.2. Steady-state Solutions
16.3. The Liouville Problem for Steady Solutions
16.4. Self-similarity
16.5. Statement of Jia and Šverák’s Theorem
16.6. The Case of Locally Bounded Initial Data
16.7. The Case of Rough Data
16.8. Non-existence of Backward Self-similar Solutions
16.9. Discretely Self-similar Solutions
16.10. Time-periodic Weak Solutions
17. α-Models
17.1. Global Existence, Uniqueness and Convergence Issues for Approximated Equations
17.2. Leray’s Mollification and the Leray-α Model
17.3. The Navier–Stokes α-Model
17.4. The Clark-α Model
17.5. The Simplified Bardina Model
17.6. Reynolds Tensor
18. Other Approximations of the Navier–Stokes Equations
18.1. Faedo–Galerkin Approximations
18.2. Frequency Cut-off
18.3. Hyperviscosity
18.4. Ladyzhenskaya’s Model
18.5. Damped Navier–Stokes Equations
19. Artificial Compressibility
19.1. Temam’s Model
19.2. Višik and Fursikov’s Model
19.3. Hyperbolic Approximation
20. Conclusion
20.1. Energy Inequalities
20.2. Critical Spaces for Mild Solutions
20.3. Models for the (Potential) Blow-up
20.4. The Method of Critical Elements
20.5. Some Open Questions
Notations and Glossary
Bibliography
Index