Turbulence for the generalised Burgers equation

TL;DR

This survey reviews turbulence results for the generalized Burgers equation on the circle, providing sharp bounds for Sobolev norms and small-scale turbulence quantities under various forcing conditions.

Contribution

It consolidates and extends previous results by establishing sharp bounds for Sobolev norms and turbulence measures in the generalized Burgers equation with different forcing scenarios.

Findings

Sharp bounds for Sobolev norms of solutions.

Confirmation of physical turbulence predictions.

Results valid for both deterministic and stochastic forcing.

Abstract

In this survey, we review the results on turbulence for the generalised Burgers equation on the circle: u_t+f'(u)u_x=\nu u_{xx}+\eta,\ 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<\nu << 1 corresponds to a viscosity coefficient. We will consider both the case \eta=0 and the case when \eta is 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 \nu^{-\delta}, \delta>=0, with the same value of \delta for upper and lower bounds, are obtained. These results yield sharp bounds for small-scale quantities characterising turbulence, confirming the physical predictions \cite{BK07}.