Collaborative Research: Branching Markov Chains and Stochastic Analysis Associated with Problems in Fluid Flow

合作研究：与流体流动问题相关的分支马尔可夫链和随机分析

• 批准号: 1408947
• 负责人: Edward Waymire
• 金额: $ 22.2万
• 依托单位: Oregon State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1408947&HistoricalAwards=false
• 关键词: Collaborative; Research; Branching; Markov; Chains

Introduced in the mid 19 century, the Navier-Stokes equations are used to analyze fluid flows from laminar to turbulent regimes. Despite their importance and usefulness in engineering and science, a complete theory establishing properties of solutions of these equations continues to be elusive. With applications to physical and biological sciences and aeronautical and naval engineering, the mechanism for energy transfer and dissipation governing fluid motions remain carefully concealed from the current methods to analyze these equations.  The dramatic improvements in efficiency attained in aircraft, naval and automotive design, serve as testimony of the economic and societal impact of improved control of basic processes modeled by these equations.In recent years problems in the study of differential equations in general, and in particularly of the Navier-Stokes equations, have given rise to interesting probabilistic structures. The current project aims to elucidate the relation between properties of solutions of the Navier-Stokes equations with properties of a class of branching Markov chains naturally associated to these equations. As well-illustrated by considering self-similar solutions of the Navier-Stokes equations, regularity properties as well as uniqueness of solutions corresponds to properties of a specific branching Markov chain in which the branching nodes have a law determined by the invariance of the Navier-Stokes equations under spatial dilation (with a corresponding time scale change) and rotations. This intrinsic branching structure motivates the formulation of an explosion problem that it is of interest in its own right from the probability point of view. It involves new considerations of the location of the left most particle of the branching process. Furthermore, the branching structure establishes a striking connection between nonlinear PDE's and branching processes that is the object of study in this proposal. A further specific objective of this proposal is to explore the consequences on the branching structure imposed by the incompressible character of the velocity field. Specifically, the Fourier transform of the solution of the Navier-Stokes equations can be represented as an expected value of a multiplicative functional defined on the nodes of the alluded branching Markov chain that reflects the incompressibility of the velocity field. An objective of this proposal is to develop the implications for energy depletion as a consequence of the algebraic structure defined by the indicated multiplication operation. Likewise, regularity and large time behavior of the solutions of the Navier-Stokes equation can also be gleaned from this representation. The proposal also involves methods of graph theory, with the use of semi-algebraic sets in the classification of nodes of random trees. Ultimately, this proposal seeks to elucidate the role of incompressibility in the multiplicative stochastic processes associated with equations of fluid flow.

Navier-Stokes方程于19世纪中期引入，用于分析从层流到动荡政权的流体流。尽管它们在工程和科学中的重要性和实用性，但建立这些股票解决方案的特性的完整理论仍然是弹性的。随着对物理和生物科学以及航空和海军工程的应用，能量转移和耗散液体运动的机制仍然仔细地隐藏在目前的方法中，以分析这些方程。飞机，海军和汽车设计中获得的效率的显着提高，可以证明改善了由这些方程式建模的基本过程的控制的经济和社会影响。近年来，在一般的微分方程研究中，尤其是Navier-Stokes方程的研究中的问题已经引起了有趣的概率结构。当前的项目旨在阐明Navier-Stokes方程解决方案的属性与一类分支的马尔可夫链的属性自然与这些方程相关的属性。通过考虑Navier-Stokes方程的自相似解决方案，规则性属性以及解决方案的唯一性对应于特定分支马尔可夫链的特性，其中分支节点具有由空间字典下Navier-Stokes方程的不变性确定的法律（具有相应的时间尺度的变化）和旋转。这种内在的分支结构激发了爆炸问题的公式，从概率的角度来看，它本身就是其权利。它涉及分支过程中左最粒子的位置的新考虑。此外，分支结构在非线性PDE和分支过程之间建立了引人注目的联系，这是该提案中研究的对象。该建议的另一个具体目标是探索速度场不可压缩特征对分支结构的后果。具体而言，Navier-Stokes方程的解的傅立叶变换可以表示为在所指的分支Markov链节点上定义的乘法功能的期望值，该函数反映了速度场的不可压缩性。该提案的一个目的是发展对能量耗竭的影响，这是由于指定的乘法操作定义的代数结构。同样，Navier-Stokes方程解决方案解决方案的规律性和较大的时间行为也可以从该表示形式中收集。该建议还涉及图理论的方法，并在随机树节点的分类中使用半代数集。最终，该提案旨在阐明与流体流动方程相关的乘法随机过程中不可压缩性的作用。