Time asymptotics for solutions of the Burgers equation with a periodic force

We consider the Burgers equation with a periodic force \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\frac{\partial}{\partial t}u+u\cdot\nabla u=\frac12\Delta u+\nabla V(x)$\end{document} which presents a simplified model for turbulence. We are interested in the asymptotic behaviour of solutions for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $t\to\infty$\end{document}. This problem has been studied by Sinai who uses a probabilistic and very technical approach. Using methods from spectral theory we get similar results. This functional analytic approach gives an easier proof. For certain initial data (periodic or some random perturbations of those) we show time-convergence towards a deterministic periodic limit solution related to the ground state of a certain Schrödinger operator.