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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 where is smooth and strongly convex, and the constant corresponds to the viscosity coefficient. Both the unforced case () and the case when is a random force which is smooth with respect to and irregular (kick or white noise) with respect to are considered. In both cases sharp bounds of the form , , are obtained for the Sobolev norms of averaged over time and over the ensemble, with the same value of for upper and lower bounds. These results yield sharp bounds for small-scale quantities characterising turbulence, confirming the physical predictions. Bibliography: 56 titles.