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

Navier-Stokes方程于19世纪中期提出，用于分析流体从层流到湍流区的流动。尽管它们在工程和科学上很重要和有用，但建立这些方程的解的性质的完整理论仍然难以捉摸。随着在物理和生物科学以及航空和海军工程中的应用，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仔细地隐藏着。在飞机、海军和汽车设计中实现的效率的显著提高，证明了改善对由这些方程建模的基本过程的控制所产生的经济和社会影响。近年来，在一般微分方程的研究中，特别是在纳维尔-斯托克斯方程的研究中，产生了有趣的概率结构。本项目旨在阐明Navier-Stokes方程的解的性质与与这些方程自然相关的一类分支马尔可夫链的性质之间的关系。通过考虑Navier-Stokes方程的自相似解，很好地说明了解的正则性和唯一性对应于特定的分枝马尔可夫链的性质，其中分枝节点具有由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爆炸问题的形成，从概率的角度来看，爆炸问题本身就是有利益的。它涉及到分支过程中最左侧粒子的位置的新考虑。此外，分支结构在非线性偏微分方程组和分支过程之间建立了显著的联系，这也是本方案的研究对象。这一建议的另一个具体目标是探讨速度场的不可压缩特性对分支结构的影响。具体地说，Navier-Stokes方程的解的傅里叶变换可以表示为定义在所暗示的分支马尔可夫链的节点上的乘法泛函的期望值，其反映了速度场的不可压缩性。这项提议的一个目标是发展由于所指示的乘法运算定义的代数结构而导致的能量耗尽的影响。同样，从这种表示也可以得到Navier-Stokes方程解的正则性和大时间性态。该建议还涉及图论的方法，使用半代数集对随机树的节点进行分类。最终，这一建议试图阐明不可压缩性在与流体流动方程相关的乘性随机过程中的作用。

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.