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

We consider a non-homogeneous generalised Burgers equation: $$\frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2} = \eta^{\omega},\quad t \in \R,\ x \in S^1.$$ Here, \nu is small and positive, f is strongly convex and satisfies a growth assumption, while \eta^{\omega} is a space-smooth random "kicked" forcing term. For any solution $u$ of this equation, we consider the quasi-stationary regime, corresponding to t>=2. After taking the ensemble average, we obtain upper estimates as well as time-averaged lower estimates for a class of Sobolev norms of $u$. 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 do not depend on the time t or the initial condition.