Dynamics near the Subcritical Transition of the 3D Couette Flow II.

Format:

Book

Author/Creator:

Bedrossian, Jacob.

Contributor:

Germain, Pierre.

Masmoudi, Nader.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.279

Language:

English

Subjects (All):

Viscous flow--Mathematical models.

Viscous flow.

Stability.

Shear flow.

Inviscid flow.

Mixing.

Damping (Mechanics).

Three-dimensional modeling.

Physical Description:

1 online resource (148 pages)

Edition:

1st ed.

Other Title:

Dynamics Near the Subcritical Transition of the 3D Couette Flow II

Place of Publication:

Providence : American Mathematical Society, 2022.

Summary:

"This is the second in a pair of works which study small disturbances to the plane, periodic 3D Couette flow in the incompressible Navier-Stokes equations at high Reynolds number Re. In this work, we show that there is constant , independent of Re, such that sufficiently regular disturbances of size for any exist at least until and in general evolve to be due to the lift-up effect. Further, after times , the streamwise dependence of the solution is rapidly diminished by a mixing-enhanced dissipation effect and the solution is attracted back to the class of "2.5 dimensional" streamwise-independent solutions (sometimes referred to as "streaks"). The largest of these streaks are expected to eventually undergo a secondary instability at . Hence, our work strongly suggests, for all (sufficiently regular) initial data, the genericity of the "lift-up effect streak growth streak breakdown" scenario for turbulent transition of the 3D Couette flow near the threshold of stability forwarded in the applied mathematics and physics literature"-- Provided by publisher.

Contents:

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

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

9781470472313

1470472317

OCLC:

1343248404