Title: Introduction to Burgers Turbulence
Abstract:
The last decades witnessed a renewal of interest in the Burgers
equation. Much activities focused on extensions of the original
one-dimensional pressureless hydrodynamical model introduced in the
thirties by the Dutch scientist J.M. Burgers, and more precisely on
the problem of Burgers turbulence, that is the study of the solutions
to the one- or multi-dimensional Burgers equation with random initial
conditions or random forcing. Such work was frequently motivated by
new emerging applications of Burgers model to statistical physics,
cosmology, and fluid dynamics. Also Burgers turbulence appeared as one
of the simplest instances of a nonlinear system out of equilibrium.
The study of random Lagrangian systems, of stochastic partial
differential equations and their invariant measures, the theory of
dynamical systems, the applications of field theory to the
understanding of dissipative anomalies and of multiscaling in
hydrodynamic turbulence have benefited significantly from progress in
Burgers turbulence. The aim of these lectures is to give a unified
view of selected work stemming from these rather diverse disciplines.
Outline of the course:
Lecture 1: Decaying Burgers equation: inviscid limit, maximum
principle and singularities; Brownian initial velocities; transport of
mass.
Lecture 2: Forced Burgers equation: variational principle, global
minimizer and topological shocks; invariant measure; extensions to
infinite systems.
Lecture 3: Kicked Burgers turbulence and connections with Aubry-Mather
theory
Lecture 4: Statistics of Burgers turbulence: shocks and bifractality;
dissipative anomaly and distributions of velocity
increments/gradients; self-similar forcing and multiscaling.
Reference:
Burgers turbulence
J. Bec & K. Khanin
Phys. Rep. 447, 1-66, 2007, [arXiv:0704.1611]