Randomness in Fluids and Waves

NSF奖摘要-DMS-0203938数学科学：流体和波中的随机性摘要Bronski流体力学是现代应用数学中发挥核心作用的一门重要学科。流体中的许多现象，如湍流和间歇，即使在物理层面上也仍然知之甚少，因此提供了丰富的数学问题来源。在这个项目中，我们将从数学上研究随机搅拌流体的输运特性。在早期的工作中，主要研究人员和合作者能够利用渐近分析、半经典特征值问题和概率的思想，推导出关于特定流体流动模型中的输运性质和诱导间歇的大量信息。在目前的工作中，我们将这些想法扩展到更复杂的流场，其中没有精确的解公式。除了上述技术，我们还打算利用Donsker-Varadhan理论，给出通过Feynman-Kac积分定义的量的大偏差原理。这个项目的目标是在流体传输领域弥合物理理解和数学证明之间的差距。目前，人们只从严格的数学意义上理解了几个非常简单的由湍流流体输送被动量(如染料)的模型。通过将数学理解扩展到更复杂的模型，我们将澄清这些简单模型具有哪些特性，而不是与科学家和工程师感兴趣的更现实的流共享的问题。我们还希望扩大对交通属性的物理理解，这还远远没有完成。

NSF Award Abstract - DMS-0203938Mathematical Sciences: Randomness in Fluids and WavesAbstract0203938 BronskiFluid mechanics is an important subject that plays a central role in modern applied mathematics. Many phenomena in fluids, such as turbulence and intermittency, are still understood poorly even at the physical level, and as such provide a rich source of mathematical problems. In this project we will study mathematically the transport properties of a randomly stirred fluid. In earlier work, the principal investigator and collaborators were able to deduce a great deal of information about the transport properties and induced intermittency in a particular model of fluid flow, using ideas of asymptotic analysis, semiclassical eigenvalue problems, and probability. In current work we extend these ideas to more complex flow fields, where exact solution formulae are not available. In addition to the above techniques, we intend to make use of the Donsker-Varadhan theory, giving large deviations principles for quantities defined via Feynman-Kac integrals.The goal of this project is to bridge the gap between physical understanding and mathematical proof in the area of fluid transport. At present only a few very simple models for the transport of a passive quantity, such as a dye, by a turbulent fluid are understood in a rigorous mathematical sense. By extending the mathematical understanding to more complex models, we will clarify the question of what properties these simple models do and do not share with the more realistic flows of interest to scientists and engineers. We also hope to expand the physical understanding of transport properties, which is still far from complete.