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.
