Stability, Mixing, and Stochastics in Hydrodynamics

描述流体流动行为的基本数学模型可以追溯到18世纪，然而，从理论的角度来看，在实验中观察到的许多现象还远远没有得到很好的理解。例如，特别具有挑战性的是当弱耗散力（例如由分子摩擦产生）与平流过程（例如混合和搅拌）相互作用时，对基本稳定机制的研究。该项目的目标是开发新的数学工具，可用于在流体动力学和流体动力学稳定性方面的未决问题上取得进展，如涡流和层流的稳定性，湍流和大气/海洋动力学中相干结构的出现，湍流的统计描述。本研究项目围绕混合和增强耗散效应的概念，通过流体，一种已被证明与动力学理论（朗道阻尼）和流体力学（无粘阻尼）中的基本耗散机制密切相关的机制。研究人员和合作者的目标是扩大非线性偏微分方程这一新兴领域，通过建立新的分析工具来研究纯数学问题（正则性问题，随机类型的扰动，无限维动力系统）和与物理现象（混合和非线性共振）广泛相关的概念之间的复杂相互作用。我们的目标是开发强大的非线性方法，从Hörmander，调和分析和奇异积分理论，概率和随机过程的意义上的hypoellipticity借用的想法。

The basic mathematical models that describe the behavior of fluid flows date back to the eighteenth century, and yet many phenomena observed in experiments are far from being well understood from a theoretical viewpoint. For instance, especially challenging is the study of fundamental stability mechanisms when weak dissipative forces (generated, for example, by molecular friction) interact with advection processes, such as mixing and stirring. The goal of this project is to develop new mathematical tools that may be used to make progress towards outstanding open problems in fluid dynamics and hydrodynamic stability, such as the stability of vortices and laminar flows, the appearance of coherent structures in turbulence and atmosphere/ocean dynamics, and the statistical description of turbulent flows.This research project revolves around the concepts of mixing and enhanced dissipation effects induced by a fluid, a mechanism that has been proven to be strongly connected to fundamental dissipation mechanisms in kinetic theory (Landau damping) and in fluid mechanics (inviscid damping). The investigator and collaborators aim to expand this emerging field of nonlinear partial differential equations, by building new analytical tools to study the complicated interaction between purely mathematical questions (regularity issues, perturbations of stochastic type, dynamical systems in infinite-dimensions) and concepts widely related to physical phenomena (mixing and nonlinear resonances). The goal is to develop robust nonlinear methods, with ideas borrowed from hypoellipticity in the sense of Hörmander, harmonic analysis and singular integral theory, and probability and stochastic processes.