6.7 Properties of the Stress Cutoff Functions

Kapitel in diesem Buch

1. Frontmatter i
2. Contents v
3. 1. Introduction
4. 1.1 Context and motivation 1
5. 1.2 Ideas and difficulties 6
6. 1.3 Organization of the book 8
7. 1.4 Acknowledgments 9
8. 2. Outline of the convex integration scheme
9. 2.1 A Guide to the Parameters 11
10. 2.2 Inductive Assumptions 14
11. 2.3 Intermittent Pipe Flows 14
12. 2.4 Higher Order Stresses 17
13. 2.5 Cutoff Functions 22
14. 2.6 The Perturbation 27
15. 2.7 The Reynolds Stress Error and Heuristic Estimates 29
16. 3. Inductive Assumptions
17. 3.1 General Notations 37
18. 3.2 Inductive Estimates 39
19. 3.3 Main Inductive Proposition 44
20. 3.4 Proof of Theorem 1.1 44
21. 4. Building Blocks
22. 4.1 A Careful Construction of Intermittent Pipe Flows 49
23. 4.2 Deformed Pipe Flows and Curved Axes 57
24. 4.3 Placements Via Relative Intermittency 59
25. 5. Mollification 67
26. 6. Cutoffs
27. 6.1 Definition of the Velocity Cutoff Functions 83
28. 6.2 Properties of the Velocity Cutoff Functions 87
29. 6.3 Definition of the Temporal Cutoff Functions 115
30. 6.4 Estimates on Flow Maps 117
31. 6.5 Stress Estimates on the Support of the New Velocity Cutoff Functions 121
32. 6.6 Definition of the Stress Cutoff Functions 123
33. 6.7 Properties of the Stress Cutoff Functions 124
34. 6.8 Definition and Properties of the Checkerboard Cutoff Functions 131
35. 6.9 Definition of the Cumulative Cutoff Function 133
36. 7. From q to q + 1: Breaking Down the Main Inductive Estimates
37. 7.1 Induction on Q 135
38. 7.2 Notations 136
39. 7.3 Induction on ñ 138
40. 8. Proving the Main Inductive Estimates
41. 8. 1 Definition of R̊_{q, n̄, p̄} and W_{q+1, n̄, p̄} 143
42. 8. 2 Estimates For W_{q+1, n̄, p̄} 147
43. 8.3 Identification of Error Terms 152
44. 8.4 Transport Errors 168
45. 8.5 Nash Errors 171
46. 8.6 Type 1 Oscillation Errors 172
47. 8.7 Type 2 Oscillation Errors 180
48. 8.8 Divergence Corrector Errors 189
49. 8.9 Time Support of Perturbations and Stresses 191
50. 9. Parameters
51. 9.1 Definitions and Hierarchy of the Parameters 193
52. 9.2 Definitions of the Q-Dependent Parameters 196
53. 9.3 Inequalities and Consequences of the Parameter Definitions 198
54. 9.4 Mollifiers and Fourier Projectors 203
55. 9.5 Notations 204
56. Appendix A: Useful Lemmas
57. Introduction 205
58. A.1 Transport Estimates 206
59. A.2 Proof of Lemma 6.2 206
60. A.3 L^p Decorrelation 209
61. A.4 Sobolev Inequality with Cutoffs 209
62. A.5 Consequences of the Faà di Bruno Formula 211
63. A.6 Bounds for Sums and Iterates of Operators 217
64. A.7 Commutators with Material Derivatives 221
65. A.8 Intermittency-Friendly Inversion of the Divergence 226
66. Bibliography 239
67. Index 245