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Navier-Stokes方程描述的，这是一个极其困难的数学研究对象。这就是为什么湍流理论也适用于在某些方面行为类似于纳维尔-斯托克斯方程动力学的模型。Burgers方程提供了一个最流行和最重要的湍流模型，近五年来，人们对紧致区域上Burgers湍流的认识有了很大的进展。这一进展是基于与随机动力系统理论的美好联系。然而，湍流理论本质上处理的是现有方法无法应用的无限域中的流体动力学。本研究的目的是在这种情况下建立Burgers湍流理论。虽然动力系统理论的方法仍将发挥重要作用，但新方法的主要新奇是基于Burgers方程和定向聚合物之间的联系。从概率论的观点来看，有向聚合物是随机环境中的随机游动。在过去的20年里，人们对这种随机游动进行了深入的研究。然而，随机势形成这样的环境中的湍流设置有一个特定的产品结构，这是以前没有研究过的数学文献。我们的目的是建立一个理论，这样的定向聚合物和使用信息的渐近性质的结构分析中的冲击Burgers湍流。拟议的研究项目基本上是跨学科的。它与偏微分方程理论、概率论、动力系统、统计力学有着密切的联系。在这个领域有许多重要的开放性问题，非常适合博士研究。我们希望有几个研究生将积极参与该项目。

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.