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，然而在实验中观察到的许多现象远远没有从理论上得到很好的理解。例如，特别具有挑战性的是研究当弱耗散力(例如，由分子摩擦产生)与平流过程(如混合和搅拌)相互作用时的基本稳定性机制。这个项目的目标是开发新的数学工具，用于解决流体动力学和流体动力稳定性中的未决问题，如涡旋和层流的稳定性，湍流和大气/海洋动力学中相干结构的出现，以及湍流的统计描述。本研究项目围绕流体诱导的混合和增强耗散效应的概念，这一机制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。研究人员和合作者的目标是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)与广泛与物理现象相关的概念(混合和非线性共振)之间的复杂相互作用，从而扩展这一新兴的非线性偏微分方程领域。目标是发展稳健的非线性方法，借鉴了Hörmander意义下的亚椭圆性、调和分析和奇异积分理论以及概率和随机过程的思想。

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.