Ergodic Theory of the Burgers Equation

The ultimate goal of these lectures is to present the ergodic theory of the Burgers equation with random force in noncompact setting. Developing such a theory has been considered a hard problem since the first publication on the randomly forced inviscid Burgers [EKMS00]. It was solved in a recent work [BCK] for the forcing of Poissonian type. The Burgers equation is a basic fluid dynamic model, and the main motivation for the study of ergodicity for Burgers equation probably comes from statistical hydrodynamics where one is interested in description of statistically steady regimes of fluid flows. It can also be interpreted as a growth model and the main idea of [BCK] is to look at the Burgers equation as a model of last passage percolation type. This allowed to use various tools from the theory of first and last passage percolation. First several sections play an introductory role. We begin with an introduction to stochastic stability in Section 2. In Section 3 we briefly discuss the progress in the ergodic theory of another important hydrodynamics model, the Navier–Stokes system with random force. Section 4 is an introduction to the Burgers equation. Section 5 is a discussion of the ergodic theory of Burgers equation with random force in compact setting. In Section 6 we introduce the Poissonian forcing for the Burgers equation. In Section 7 we state the ergodic results from [Bak12] on quasicompact setting. In Section 8 we state the main results and the proof is given in Sections 9 through 13. In Section 14 we give some concluding remarks. Although many of the proofs given here are detailed and rigorous, often we give only the ideas behind the proof referring the reader to the details in [BCK].