Turbulence for the generalised Burgers equation

This survey reviews rigorous results obtained by A. Biryuk and the author on turbulence for the generalised space-periodic Burgers equation
$$ u_t+f'(u)u_x=\nu u_{xx}+\eta,\qquad x \in S^1=\mathbb{R}/\mathbb{Z}, $$
where $f$ is smooth and strongly convex, and the constant $0<\nu\ll 1$ corresponds to the viscosity coefficient. Both the unforced case ($\eta=0$) and the case when $\eta$ is a random force which is smooth with respect to $x$ and irregular (kick or white noise) with respect to $t$ are considered. In both cases sharp bounds of the form $C\nu^{-\delta}$, $\delta\geqslant 0$, are obtained for the Sobolev norms of $u$ averaged over time and over the ensemble, with the same value of $\delta$ for upper and lower bounds. These results yield sharp bounds for small-scale quantities characterising turbulence, confirming the physical predictions.
Bibliography: 56 titles.