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Product Code:  SURV/255 
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eBook ISBN:  9781470465643 
Product Code:  SURV/255.E 
List Price:  $125.00 
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Softcover ISBN:  9781470464363 
eBook: ISBN:  9781470465643 
Product Code:  SURV/255.B 
List Price:  $250.00 $187.50 
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Softcover ISBN:  9781470464363 
Product Code:  SURV/255 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
eBook ISBN:  9781470465643 
Product Code:  SURV/255.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Softcover ISBN:  9781470464363 
eBook ISBN:  9781470465643 
Product Code:  SURV/255.B 
List Price:  $250.00 $187.50 
MAA Member Price:  $225.00 $168.75 
AMS Member Price:  $200.00 $150.00 

Book DetailsMathematical Surveys and MonographsVolume: 255; 2021; 192 ppMSC: Primary 60; Secondary 35; 37; 76
This book is dedicated to the qualitative theory of the stochastic onedimensional Burgers equation with small viscosity under periodic boundary conditions and to interpreting the obtained results in terms of onedimensional turbulence in a fictitious onedimensional fluid described by the Burgers equation. The properties of onedimensional 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 onedimensional 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 selfconsistent 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 onedimensional 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.
ReadershipGraduate students and researchers interested in stochastic differential equations and turbulence.

Table of Contents

Chapters

Introduction

Stochastic Burgers equation

Basic results

Asymptotically sharp estimates for Sobolev norms of solutions

Mixing in the stochastic Burgers equation

Stochastic Burgers equation in the space $L_1$

Notes and comments, I

Onedimensional turbulence

Turbulence and burgulence

Rigorous burgulence

The inviscid limit and inviscid burgulence

Notes and comments, II

Additional material

Miscellanea

Appendices

Solutions for selected exercises


Additional Material

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This book is dedicated to the qualitative theory of the stochastic onedimensional Burgers equation with small viscosity under periodic boundary conditions and to interpreting the obtained results in terms of onedimensional turbulence in a fictitious onedimensional fluid described by the Burgers equation. The properties of onedimensional 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 onedimensional 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 selfconsistent 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 onedimensional 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.
Graduate students and researchers interested in stochastic differential equations and turbulence.

Chapters

Introduction

Stochastic Burgers equation

Basic results

Asymptotically sharp estimates for Sobolev norms of solutions

Mixing in the stochastic Burgers equation

Stochastic Burgers equation in the space $L_1$

Notes and comments, I

Onedimensional turbulence

Turbulence and burgulence

Rigorous burgulence

The inviscid limit and inviscid burgulence

Notes and comments, II

Additional material

Miscellanea

Appendices

Solutions for selected exercises