Time asymptotics for solutions of the Burgers equation with a periodic force (Q1961438)

The paper considers the Burgers equation \({\partial\over\partial t}u+u\cdot\nabla u={1\over 2}\Delta u+\nabla V(x)\), \(t\geq 0\), \(x\in \mathbb{R}^d\), with initial data \(u(0,x)=u_0(x)\), serving as a simplified model of turbulent fluid motion. The problem to investigate is the behaviour of the solutions as \(t\to\infty\). This problem had been studied by \textit{Ya. G. Sinai} [J. Stat. Phys. 64, 1-12 (1991)] with probabilistic methods. The main result was of the type that if the potential \(V\) is periodic then almost surely the solution of the Burgers equation tends to a periodic limit solution related to the ground state of a certain Schrödinger operator, which does not depend on the initial value. The present paper states similar results using methods from spectral theory instead of probabilistic ones.