Title: One-Dimensional Turbulence and the Stochastic Burgers Equation
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 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.
Subjects: Mathematics, Stochastic partial differential equations, Burgers equation, Turbulence, Mathematical models, Probability theory and stochastic processes {For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX} -- Stochastic analysis [See also 58J65] -- Stochastic partial differentia, Partial differential equations -- Hyperbolic equations and systems [See also 58J45] -- Conservation laws, Partial differential equations -- Equations of mathematical physics and other areas of application [See also 35J05, 35J10, 35K05, 35L05] -- PDEs in connection with fluid mechanics, Dynamical systems and ergodic theory [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-XX] -- Ergodic theory [See also 28Dxx] -- Ergodicity, mixing, rates of mixing, Fluid mechanics {For general continuum mechanics, see 74Axx, or other parts of 74-XX} -- Turbulence [See also 37-XX, 60Gxx, 60Jxx] -- Fundamentals, Fluid mechanics {For general continuum mechanics, see 74Axx, or other parts of 74-XX} -- Turbulence [See also 37-XX, 60Gxx, 60Jxx] -- Turbulent transport, mixing