Bedrossian, Germain, and Manmoudi complete their two-volume study of small disturbances to the plane, periodic three-dimensional Couette flow in the incompressible Navier-Stokes equations at high Reynolds number. Their findings strongly suggest, for all (sufficiently regular) initial data, the genericity of the "lift-up effect to streak growth to streak breakdown" scenario for turbulent transition of the three-dimensional Couette flow near the threshold of stability forwarded in the applied mathematics and physics literature. Annotation ©2022 Ringgold, Inc., Portland, OR (protoview.com)
Author(s): Jacob Bedrossian, Pierre Germain, Nader Masmoudi
Series: Memoirs of the American Mathematical Society, 1377
Publisher: American Mathematical Society
Year: 2022
Language: English
Pages: 147
City: Providence
Cover
Title page
Chapter 1. Introduction
1.1. Linear behavior and streaks
1.2. Statement of main results
1.3. Notations and conventions
Acknowledgments
Chapter 2. Outline of the proof
2.1. Summary and weakly nonlinear heuristics
2.2. Choice of the norms
2.3. Instantaneous regularization and continuation of solutions
2.4. ?ⁱ formulation, the coordinate transformation, and some key cancellations
2.5. The toy model and design of the norms
2.6. Design of the norms based on the toy model
2.7. Main energy estimates
Chapter 3. Regularization and continuation
Chapter 4. Multiplier and paraproduct tools
4.1. Basic inequalities regarding the multipliers
4.2. Paraproducts and related notations
4.3. Product lemmas and a few immediate consequences
Chapter 5. High norm estimate on ?²
5.1. Zero frequencies
5.2. Non-zero frequencies
Chapter 6. High norm estimate on ?³
6.1. Zero frequencies
6.2. Non-zero frequencies
Chapter 7. High norm estimate on ?¹₀
7.1. Transport nonlinearity
7.2. Nonlinear stretching
7.3. Forcing from non-zero frequencies
7.4. Dissipation error terms
Chapter 8. High norm estimate on ?¹_{≠}
8.1. Linear stretching term ??1
8.2. Lift-up effect term ??
8.3. Linear pressure term ??1
8.4. Nonlinear pressure ???
8.5. Nonlinear stretching ???
8.6. Transport nonlinearity ?
8.7. Dissipation error terms ?
Chapter 9. Coordinate system controls
9.1. High norm estimate on ?
9.2. Low norm estimate on ?
9.3. Long time, high norm estimate on ?ⁱ
9.4. Shorter time, high norm estimate on ?ⁱ
9.5. Low norm estimate on ?
Chapter 10. Enhanced dissipation estimates
10.1. Enhanced dissipation of ?³
10.2. Enhanced dissipation of ?²
10.3. Enhanced dissipation of ?¹
Chapter 11. Sobolev estimates
11.1. Improvement of (2.45c)and (2.45b)
11.2. Improvement of (2.45a)
Appendix A. Fourier analysis conventions, elementary inequalities, and Gevrey spaces
Appendix B. Some details regarding the coordinate transform
Appendix C. Definition and analysis of the norms
C.1. Definition and analysis of ?
C.2. The design and analysis of ?_{?}
Appendix D. Elliptic estimates
D.1. Lossy estimates
D.2. Precision lemmas
Bibliography
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