**Mathematical Surveys and Monographs**

Volume: 255;
2021;
192 pp;
Softcover

MSC: Primary 60;
Secondary 35; 37; 76

**Print ISBN: 978-1-4704-6436-3
Product Code: SURV/255**

List Price: $125.00

AMS Member Price: $100.00

MAA Member Price: $112.50

**Electronic ISBN: 978-1-4704-6564-3
Product Code: SURV/255.E**

List Price: $125.00

AMS Member Price: $100.00

MAA Member Price: $112.50

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#### Supplemental Materials

# One-Dimensional Turbulence and the Stochastic Burgers Equation

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*Alexandre Boritchev; Sergei Kuksin*

This book is dedicated to the qualitative
theory of the stochastic one-dimensional Burgers equation with small
viscosity under periodic boundary conditions and to interpreting the
obtained results in terms of one-dimensional turbulence in a
fictitious one-dimensional fluid described by the Burgers
equation. The properties of one-dimensional turbulence which we
rigorously derive are then compared with the heuristic Kolmogorov
theory of hydrodynamical turbulence, known as the K41 theory. It is
shown, in particular, that these properties imply natural
one-dimensional analogues of three principal laws of the K41 theory:
the size of the Kolmogorov inner scale, the \(2/3\)-law, and the
Kolmogorov–Obukhov law.

The first part of the book deals with the stochastic Burgers
equation, including the inviscid limit for the equation, its
asymptotic in time behavior, and a theory of generalised
\(L_1\)-solutions. This section makes a self-consistent
introduction to stochastic PDEs. The relative simplicity of the model
allows us to present in a light form many of the main ideas from the
general theory of this field. The second part, dedicated to the
relation of one-dimensional turbulence with the K41 theory, could
serve for a mathematical reader as a rigorous introduction to the
literature on hydrodynamical turbulence, all of which is written on a
physical level of rigor.

#### Readership

Graduate students and researchers interested in stochastic differential equations and turbulence.

#### Table of Contents

# Table of Contents

## One-Dimensional Turbulence and the Stochastic Burgers Equation

- Cover Cover11
- Title page iii4
- Introduction 110
- Part 1. Stochastic Burgers Equation 1120
- Chapter 1. Basic results 1322
- 1.1. Introduction 1322
- 1.2. Properties of the random force 𝜉 1524
- 1.3. Deterministic Cauchy problem 2029
- \altindent Preliminaries 2029
- \altindent Well-posedness of equation (B) 2332
- 1.4. Strong solutions of the stochastic Cauchy problem 3039
- \altindent Notion of a strong solution and its existence 3039
- \altindent Basic estimates for strong solutions 3140
- \altindent The balance of energy relation 3443
- \altindent The vorticity stretching 3443
- 1.5. Weak solutions and space–homogeneous solutions 3544
- \altindent Weak solutions of (B) 3544
- \altindent Space-homogeneous random fields and solutions 3645
- 1.6. The two Markov semigroups and the Markov property 3746
- \altindent The semigroup for measures 3746
- \altindent The semigroup for functions 3948
- \altindent Kolmogorov-Chapman relation and Markov property 4049
- 1.7. Weak convergence of measures 4251
- 1.8. Appendix: More on the deterministic Burgers equation 4453
- \altindent Analyticity of the flow of the deterministic equation (B) 4453
- \altindent Burgers equation with a regular force 4655

- Chapter 2. Asymptotically sharp estimates for Sobolev norms of solutions 4958
- 2.1. Oleinik’s estimate 4958
- \altindent A version of Oleinik’s estimate 5261
- 2.2. Upper bounds for moments of Sobolev norms of solutions 5362
- 2.3. Energy balance and lower bounds for moments of norms of solutions 5665
- \altindent Bounds for moments of 𝐻^{𝑚} norms, 𝑚≥1 5665
- \altindent Bounds for moments of 𝑊^{𝑚}_{𝑝} norms 6069
- \altindent Bounds for moments of 𝐿_{𝑝} norms 6271

- Chapter 3. Mixing in the stochastic Burgers equation 6574
- Chapter 4. Stochastic Burgers equation in the space 𝐿₁ 7786
- Chapter 5. Notes and comments, I 8594

- Part 2. One-Dimensional Turbulence 8796
- Chapter 6. Turbulence and Burgulence 8998
- Chapter 7. Rigorous burgulence 101110
- 7.1. Dissipation scale 102111
- \altindent Energy and inertial ranges 103112
- 7.2. Structure function and intermittency 104113
- \altindent Structure function 104113
- \altindent Dissipation range in the 𝑥-presentation 108117
- \altindent Intermittency 109118
- 7.3. Energy spectrum 110119
- 7.4. The limiting in time behaviour 112121
- \altindent The energy spectrum 112121
- \altindent The structure function 113122
- 7.5. Appendix: High–probability versions of the results above 114123

- Chapter 8. The Inviscid limit and Inviscid Burgulence 119128
- 8.1. Cauchy problem for an auxiliary deterministic equation 119128
- 8.2. Inviscid limit for the deterministic equation 121130
- 8.3. Inviscid limit for the stochastic equation 124133
- \altindent The limiting process in 𝐿₁ 125134
- 8.4. Inviscid Burgulence 127136
- 8.5. Mixing for the entropy solutions 128137
- \altindent Inviscid limit for the stationary measure for eq. (1.1.12) 128137
- \altindent Mixing for the entropy solutions 129138
- \altindent Time-asymptotic for the inviscid energy spectrum and structure function 129138

- Chapter 9. Notes and comments, II 133142

- Part 3. Additional Material 135144
- Chapter 10. Miscellanea 137146
- 10.1. Other types of forces: Estimates, Burgulence and mixing 137146
- 10.2. High–frequency kick forces 143152
- 10.3. Exponential mixing and its consequences 151160
- \altindent Exponential mixing in the Burgers equation with kick-forces and red forces 151160
- \altindent Exponential mixing and ergodicity 151160
- 10.4. Other equations 153162

- Chapter 11. Appendices 157166
- 11.1. Appendix A: Preliminaries from analysis 157166
- 11.2. Appendix B: Spaces of functions 160169
- 11.3. Appendix C: Measurable spaces, probability spaces and transition probabilities 162171
- 11.4. Appendix D: Kantorovich distance 166175
- 11.5. Appendix E: Random processes and the Wiener process 167176
- 11.6. Appendix F: Filtered probability spaces and Markov processes 169178
- 11.7. Appendix G: Stopping and hitting times and the strong Markov property 170179
- 11.8. Appendix H: Stochastic differential equations and Ito’s formula 171180

- Chapter 12. Solutions for selected exercises 175184
- Acknowledgements 181190
- Bibliography 183192
- Index 191200

- Back Cover Back Cover1203