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

在世纪中期引入的Navier-Stokes方程用于分析从层流到湍流的流体流动。尽管它们在工程和科学中的重要性和实用性，建立这些方程的解的性质的完整理论仍然是难以捉摸的。随着物理和生物科学以及航空和海军工程的应用，控制流体运动的能量传递和耗散机制仍然被当前分析这些方程的方法小心地隐藏。 在飞机、海军和汽车设计中所获得的效率的显著提高，证明了用这些方程模拟的基本过程的改进控制的经济和社会影响。近年来，一般微分方程，特别是Navier-Stokes方程的研究中的问题，引起了有趣的概率结构。目前的项目旨在阐明Navier-Stokes方程的解的性质与自然地与这些方程相关联的一类分支马尔可夫链的性质之间的关系。 正如通过考虑Navier-Stokes方程的自相似解很好地说明的那样，解的正则性以及唯一性对应于特定分支马尔可夫链的性质，其中分支节点具有由Navier-Stokes方程在空间膨胀（具有相应的时间尺度变化）和旋转下的不变性确定的规律。 这种内在的分支结构激发了一个爆炸问题的公式化，从概率的角度来看，它本身就是一个有趣的问题。它涉及对分支过程最左边粒子位置的新考虑。此外，分支结构建立了一个显着的非线性偏微分方程和分支过程之间的联系，这是在这个建议的研究对象。这个建议的另一个具体目标是探索由不可压缩特性的速度场所施加的分支结构的后果。 具体地说，Navier-Stokes方程的解的傅里叶变换可以表示为在所提到的分支马尔可夫链的节点上定义的乘法泛函的期望值，其反映了速度场的不可压缩性。 本建议的一个目标是发展的影响，能源消耗的代数结构所定义的乘法运算的结果。同样，Navier-Stokes方程的解的规律性和大时间行为也可以从这个表示中收集。 该提案还涉及图论的方法，使用半代数集的节点分类的随机树。 最后，这个建议旨在阐明不可压缩性的作用与流体流动方程的乘法随机过程。

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.