1304.6814 (Alexandre Boritchev)

Turbulence for the generalised Burgers equation    [PDF]

Alexandre Boritchev

In this survey, we review the results obtained 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 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 \geq 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}.

View original: http://arxiv.org/abs/1304.6814