One-Dimensional Turbulence and the Stochastic Burgers Equation

Alexandre Boritchev, Sergei Kuksin (Autoren)

Buch | Softcover

192 Seiten

2021
American Mathematical Society (Verlag)
978-1-4704-6436-3 (ISBN)

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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.

Alexandre Boritchev, Universite Claude Bernard Lyon 1, Villeurbanne, France. Sergei Kuksin, Universite Paris-Diderot, France, and Shandong University, Jinan, People's Republic of China.

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

Erscheinungsdatum	01.10.2021
Reihe/Serie	Mathematical Surveys and Monographs
Verlagsort	Providence
Sprache	englisch
Maße	178 x 254 mm
Gewicht	369 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
ISBN-10	1-4704-6436-5 / 1470464365
ISBN-13	978-1-4704-6436-3 / 9781470464363
Zustand	Neuware