1201.5567 (Alexandre Boritchev)

Sharp Estimates for Turbulence in White-Forced Generalised Burgers
Equation    [PDF]

Alexandre Boritchev

We consider a non-homogeneous generalised Burgers equation
\frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - \nu
\frac{\partial^2 u}{\partial x^2} = \eta,\ t \geq 0,\ x \in S^1.
Here $f$ is strongly convex, $\nu$ is small and positive, while $\eta$ is a
random forcing term, smooth in space and white in time. For any solution $u$ of
this equation we consider the quasi-stationary regime, corresponding to $t \geq
T_1$, where $T_1$ depends only on $f$ and on the distribution of $\eta$. We
obtain sharp upper and lower bounds for Sobolev norms of $u$ averaged in time
and in ensemble. These bounds imply, in their turn, sharp upper and lower
bounds for natural analogues of quantities characterising the hydrodynamical
turbulence. This gives us important information on the Burgulence, that is the
turbulence for the Burgers equation. Similar estimates have been obtained in a
similar setting in \cite{AFLV} on a physical level of rigour; we use some
arguments from that article. All our estimates do not depend on the initial
condition, and hold uniformly in $\nu$.

View original: http://arxiv.org/abs/1201.5567