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Turbulence for the generalised Burgers equation

In this survey, we review the results on turbulence for the generalised Burgers equation on the circle: u_t+f'(u)u_x=νu_{xx}+η,\ x \in S^1=\R/\Z, obtained by A.Biryuk and the author in \cite{Bir01,BorK,BorW,BorD}. Here, f is smooth and strongly convex, whereas the constant 0<ν<< 1 corresponds to a viscosity coefficient. We will consider both the case η=0 and the case when ηis a random force which is smooth in x and irregular (kick or white noise) in t. In both cases, sharp bounds for Sobolev norms of u averaged in time and in ensemble of the type C ν^{-δ}, δ>=0, with the same value of δfor upper and lower bounds, are obtained. These results yield sharp bounds for small-scale quantities characterising turbulence, confirming the physical predictions \cite{BK07}.