Physics – Mathematical Physics
Scientific paper
2012-01-26
Physics
Mathematical Physics
Scientific paper
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 results yield sharp upper and lower bounds for natural analogues of quantities characterising the hydrodynamical turbulence. All our bounds do not depend on the initial condition, and hold uniformly in $\nu$. Estimates similar to some of our results have been obtained by Aurell, Frisch, Lutsko, and Vergassola on a physical level of rigour; we use some arguments from their article.
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