Integrability and turbulence

该项目的主要目标是发现数学结果，解释在物理科学中具有根本重要性的几个模型中随机性的形成和传播。这种现象最常见的例子是三维流动中的紊流。该项目还包括一维非线性波的研究和基本数值算法的随机性传播。这些例子比充分发展的湍流更容易处理，而且也是有内在意义的。本计画的理论成果有助于发展不确定性量化的数值方法，其中包括三个计画：（a）一维精确可解模型的研究，包括标量守恒律与可积偏微分方程;（B）广泛使用的迭代特征值演算法的执行时间统计;（c）不可压缩流体中各向同性均匀紊流的随机流体流动模拟。项目（a）和（B）的目的是利用可积系统和概率论的方法来描述一类新的精确可解随机模型。项目（c）的目的是根据实验观察到的标度律，对描述充分发展的湍流的不可压缩流的欧拉方程进行求解。

The main goal of this project is to discover mathematical results that explain the formation and propagation of randomness in several models of fundamental importance in the physical sciences. The most familiar example of such phenomena is turbulence in three-dimensional flows. The project also includes the study of nonlinear waves in one dimension and the propagation of randomness by basic numerical algorithms. These examples are more tractable than fully developed turbulence, and are also of intrinsic interest. The theoretical results developed in this project assist the development of numerical methods for uncertainty quantification.Three projects are considered: (a) a study of exactly solvable models in one-dimension including scalar conservation laws and integrable partial differential equations; (b) run time statistics for widely used iterative eigenvalue algorithms; and (c) random fluid flows modeling isotropic homogeneous turbulence in incompressible fluids. The purpose of projects (a) and (b) is to use the methods of integrable systems and probability theory to describe new classes of exactly solvable stochastic models. The purpose of project (c) is to develop solutions to the Euler equations of incompressible flow that describe fully developed turbulence, in consonance with experimentally observed scaling laws.