Directed polymers and burgers turbulence.

湍流问题是现代数学和物理学中最重要和最具挑战性的开放性问题之一。流体动力学是用纳维-斯托克斯系统来描述的，这是数学研究的一个极其困难的对象。这就是为什么湍流理论也适用于某些行为类似于纳维-斯托克斯方程动力学的模型。最流行和最重要的模型之一是由汉堡方程提供的。在过去的五年中，人们对紧域上的Burgers湍流的理解取得了很大的进展。这一进展是基于与随机动力系统理论的优美联系。然而，湍流理论本质上处理的是无限域中的流体动力学，而现有的方法是无法应用的。提出的研究项目旨在构建这种情况下的Burgers湍流理论。虽然动力系统理论的方法仍将发挥重要作用，但新方法的主要新颖之处是基于Burgers方程和定向聚合物之间的联系。从概率论的角度看，定向聚合物是在随机环境中的随机游走。在过去的20年里，人们对这种随机漫步进行了深入研究。然而，在湍流环境中形成的随机势具有特定的积结构，这在以前的数学文献中没有研究过。我们的目的是建立这种定向聚合物的理论，并利用其渐近性质的信息来分析Burgers湍流中的冲击结构。拟议的研究项目本质上是跨学科的。它与PDE理论，概率论，动力系统，统计力学密切相关。这一领域有许多重要的开放问题，非常适合博士研究。我们期望有几个研究生会积极参与这个项目。

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.