Turbulence for the generalised Burgers equation

A. A. Boritchev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 69 (2014) no. 6, pp. 957-994

Résumé

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.

Détail

Keywords: Burgers equation, stochastic partial differential equations, intermittency, stationary measure.
Mots-clés : turbulence