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

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

2011-07-25

arxiv.org/abs/1107.4866v1

Mathematics

Analysis of PDEs

Scientific paper

We consider a non-homogeneous generalised Burgers equation: $$ \frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2} = \eta^{\omega},\quad t \in \R,\ x \in S^1. $$ 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 $t \geq 2$. 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.

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