One-dimensional turbulence and the stochastic Burgers equation

Author(s)

• Boritchev, Alexandre
• Kuksin, Sergej B.

Bibliographic Information

One-dimensional turbulence and the stochastic Burgers equation

Alexandre Boritchev, Sergei Kuksin

（Mathematical surveys and monographs, v. 255）

American Mathematical Society, c2021

• : pbk

Note

Includes bibliographical references (p. 183-190) and index

Description and Table of Contents

Description

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.

Table of Contents

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 One-dimensional turbulence: Turbulence and burgulence Rigorous burgulence The inviscid limit and inviscid burgulence Notes and comments, II Additional material: Miscellanea Appendices Solutions for selected exercises Acknowledgements Bibliography Index

by "Nielsen BookData"

Details

• NCID: BC0843181X
• ISBN:
  • 9781470464363
• LCCN: 2021004451
• Country Code: us
• Title Language Code: eng
• Text Language Code: eng
• Place of Publication: Providence, R.I.
• Pages/Volumes: vii, 192 p.
• Size: 26 cm
• Classification:
  • LCC : QA274.25
  • DC23 : 515/.353
  • DC23 : 515.353
• Subject Headings:
  • LCSH : Stochastic partial differential equations
  • LCSH : Burgers equation
  • LCSH : Turbulence -- Mathematical models
• Parent Bibliography ID:
  • BA00134419