On backward-time behaviour of Burgers' original model for turbulence

Burgers' model for turbulence consists of a PDE coupled with a nonlocal ODE. We study the behaviour, backward in time, of its solutions, which turns out to be quite different from the behaviour of the solutions of the space periodic two-dimensional Navier-Stokes equations (Constantin P, Foias C, Kukavica I and Majda A 1997 Dirichlet quotients and 2-D periodic Navier-Stokes equations J. Math. Pure Appl. 76 125-53) and space periodic one-dimensional Kuramoto-Sivashinsky equation (Kukavica 11992 On the behavior of the solutions of the Kuramoto-Sivashinsky equations for negative time J. Math. Anal. Appl. 166 601-6).