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We consider a non-homogeneous generalised Burgers equation: $$ \frac{\partialu}{\partial t} + f'(u)\frac{\partial u}{\partial x} - \nu \frac{\partial^2u}{\partial x^2} = \eta^{\omega},\quad t \in \R,\ x \in S^1. $$ Here, \nu issmall and positive, f is strongly convex and satisfies a growth assumption,while \eta^{\omega} is a space-smooth random "kicked" forcing term. For anysolution $u$ of this equation, we consider the quasi-stationary regime,corresponding to t>=2. After taking the ensemble average, we obtain upperestimates as well as time-averaged lower estimates for a class of Sobolev normsof $u$. These estimates are of the form C \nu^{-\beta} with the same values of$\beta$ for bounds from above and from below. They depend on \eta and f, but donot depend on the time t or the initial condition.

Estimates for Solutions of a Low-Viscosity Kick-Forced Generalised Burgers Equation