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Softcover ISBN:  9781470442170 
Product Code:  MEMO/266/1294 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
eBook ISBN:  9781470462512 
Product Code:  MEMO/266/1294.E 
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MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Softcover ISBN:  9781470442170 
eBook ISBN:  9781470462512 
Product Code:  MEMO/266/1294.B 
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MAA Member Price:  $153.00 $114.75 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 266; 2020; 154 ppMSC: Primary 35; 76
The authors study small disturbances to the periodic, plane Couette flow in the 3D incompressible NavierStokes equations at high Reynolds number
Re . They prove that for sufficiently regular initial data of size \(\epsilon \leq c_0\mathbf {Re}^1\) for some universal \(c_0 > 0\), the solution is global, remains within \(O(c_0)\) of the Couette flow in \(L^2\), and returns to the Couette flow as \(t \rightarrow \infty \). For times \(t \gtrsim \mathbf {Re}^1/3\), the streamwise dependence is damped by a mixingenhanced dissipation effect and the solution is rapidly attracted to the class of “2.5 dimensional” streamwiseindependent solutions referred to as streaks. 
Table of Contents

Chapters

1. Introduction

2. Outline of the proof

3. Regularization and continuation

4. High norm estimate on $Q^2$

5. High norm estimate on $Q^3$

6. High norm estimate on $Q^1_0$

7. High norm estimate on $Q^1_{\neq }$

8. Coordinate system controls

9. Enhanced dissipation estimates

10. Sobolev estimates

Acknowledgments

A. Fourier analysis conventions, elementary inequalities, and Gevrey spaces

B. Definition and analysis of norms

C. Multiplier and paraproduct tools

D. Elliptic estimates


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The authors study small disturbances to the periodic, plane Couette flow in the 3D incompressible NavierStokes equations at high Reynolds number

Chapters

1. Introduction

2. Outline of the proof

3. Regularization and continuation

4. High norm estimate on $Q^2$

5. High norm estimate on $Q^3$

6. High norm estimate on $Q^1_0$

7. High norm estimate on $Q^1_{\neq }$

8. Coordinate system controls

9. Enhanced dissipation estimates

10. Sobolev estimates

Acknowledgments

A. Fourier analysis conventions, elementary inequalities, and Gevrey spaces

B. Definition and analysis of norms

C. Multiplier and paraproduct tools

D. Elliptic estimates