Integrability and turbulence

这个项目的主要目标是发现数学结果，以解释在物理科学中基本重要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湍流。该项目还包括用基本的数值算法研究一维非线性波和随机性的传播。这些例子比充分发展的湍流更容易处理，而且也具有内在的意义。这个项目的理论结果有助于不确定性量化的数值方法的发展。(A)一维精确可解模型的研究，包括标量守恒律和可积偏微分方程组；(B)广泛使用的迭代特征值算法的运行时间统计；(C)模拟不可压缩流体中各向同性均匀湍流的随机流体流动。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的随机模型。项目(C)的目的是开发描述充分发展的湍流的不可压缩流动的欧拉方程的解，与实验观察到的标度定律一致。

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.