Randomness in Fluids and Waves

美国国家科学基金会奖摘要- DMS-0203938数学科学：流体和波的随机性摘要0203938布朗斯基流体力学是一门重要的学科，在现代应用数学中起着核心作用。 流体中的许多现象，如湍流和不稳定性，即使在物理水平上也仍然知之甚少，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，主要研究者和合作者能够利用渐近分析、半经典本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不稳定性的大量信息。 在目前的工作中，我们将这些想法扩展到更复杂的流场，精确的解公式是不可用的。 除了上述技术外，我们还打算利用Donsker-Varadhan理论，通过Feynman-Kac积分给出定义的量的大偏差原理。该项目的目标是弥合流体传输领域物理理解和数学证明之间的差距。 目前，只有几个非常简单的模型的被动量，如染料，通过湍流流体的传输是在严格的数学意义上理解。 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与科学家和工程师感兴趣的更现实的流动具有哪些特性。 我们还希望扩大对输运性质的物理理解，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.