Directed polymers and burgers turbulence.

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The problem of turbulence is one of the most important and challenging open problems in modern mathematics and physics. The dynamics of fluids is described by Navier-Stokes system which is an extremely difficult object for mathematical studies. That is why turbulence theory operates also with models whose behaviour in certain respects is similar to dynamics of Navier-Stokes equations. One of the most popular and important models is provided by the Burgers equation.The last five years have seen a big progress in understanding of Burgers turbulence on compact domains. This progress was based on beautiful connections with the theory of random dynamical systems. However, the turbulence theory essentially deals with dynamics of fluids in infinite domains where existing approaches cannot be applied. The proposed research project aims to construct the theory of Burgers turbulence in such a situation. While the methods of the theory of dynamical systems will still play an important role, the main novelty of the new approach is based on connections between Burgers equation and directed polymers. From the point of view of probability theory directed polymers are  random walks in random environment. Such random walks were intensively studied over the last 20 years. However, random potentials forming such an environment in the turbulence setting have a particular product structure which was not studied previously in the mathematical literature. Our aim is to build up a theory of such directed polymers and use information on their asymptotic properties in the analysis of the structure of shocks in Burgers turbulence. The proposed research project is essentially interdisciplinary. It is strongly connected with the theory of PDE's, probability theory, dynamical systems, statistical mechanics. There are many important open problems in this area which are well suited for PhD studies. We expect that several graduate students will be actively involved in the project.