Includes bibliographical references (pages 183-190) and index.
Contents:
Basic results -- Asymptotically sharp estimates for Sobolev norms of solutions -- Mixing in the stochastic Burgers equation -- Stochastic Burgers equation in the space L₁ -- Notes and comments, I -- Turbulence and burgulence -- Rigorous burgulence -- The inviscid limit and inviscid burgulence -- Notes and comments, II -- Miscellanea -- Appendices -- Solutions for selected exercises.
Summary:
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 L₁-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. Publisher.
Series:
Mathematical surveys and monographs, 0076-5376 ; volume 255
This resource is supported by the Institute of Museum and Library Services under the provisions of the Library Services and Technology Act as administered by State Library of Iowa.