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

Abstract

We consider a non-homogeneous generalised Burgers equation: $$\frac{\partial u}{\partial t}+f^{\prime }(u)\frac{\partial u}{\partial x}-\nu \frac{{\partial }^{2}u}{\partial x^2}={\eta }^{\omega },t\in \mathbb{R},x\in S^1.$$ Here, νis small and positive, f is strongly convex and satisfies a growth assumption, while η^ω 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 ν^{-β} with the same values of $\beta$ for bounds from above and from below. They depend on ηand f, but do not depend on the time t or the initial condition.

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