Ergodic and Statistical Phenomena in Dynamics

Pi：Yakov Sinai，普林斯顿大学DMS-0245397摘要：研究将朝着三个不同的方向进行：1)动力系统理论的方法分析流体动力学方程的解：3D-Navier-Stokes系统的存在性问题仍然是一个挑战。动力系统理论可以为研究(3x+1)-问题和其他数论动力系统找到新的方向：这些问题是长期存在的和容易表述的问题。其目的是找出隐含的结构，以帮助在典型轨道的分析中取得进展。3)绝热活塞问题：它是非平衡统计力学中的热门问题之一。遍历理论和概率论的方法是有用的。动力系统理论是数学的一个交叉分支，在物理、工程、生物等领域有着广泛的应用。人们认为它将应用于流体力学、数论和非平衡统计力学中的一些问题。在所有情况下，我都打算在每一种情况下解决几个核心问题。

PI: Yakov Sinai, Princeton UniversityDMS-0245397Abstract:The research will go in three different directions:1] Methods of the theory of dynamical systems for the analysis of solutions of equations of hydrodynamics: The existence problem for the 3D-Navier-Stokes system remains a challenge. The theory of dynamical systems can help to find new directions for research.2] (3x+1) - problem and other number-theoretic dynamical systems: Theseare long-standing and easily formulated problems. The goal is to find hidden structures which can help to get a progress in the analysis of typical orbits.3] Problem of adiabatic piston: It is one of popular problems in non-equilibrium statistical mechanics. Methods from ergodic theory and probability theory can be useful.The theory if dynamical systems is an interdisciplinary branch of mathematics which has many applications in physics, engineering, biology, etc. It is assumed that it will be applied to some problems in hydrodynamics, number theory and non-equilibrium statistical mechanics. In all cases I intend to work on several central problems in each of these cases.