Instability, Chaos, and Mixing in Stochastic Fluid Mechanics and Related Models

这个项目涉及与湍流运动有关的模型的研究。紊流的一个决定性特征是其混乱的、看似不可预测的行为。一种被广泛认可的描述这种混沌行为的方法是表现出相对于初始条件的极端指数灵敏度，通常被称为“蝴蝶效应”，即流体状态的微小变化在短时间内导致非常大的变化。尽管它具有基本的性质，但很少有数学工具可用于严格验证给定系统中的指数灵敏度，特别是在湍流系统中。本项目旨在开发数学工具来证明在存在少量噪声的情况下流体力学中各种模型的这种敏感性。这种噪声通常用于模拟不可预测的环境影响或粒子的小规模波动的影响。这个项目的目标是获得新的见解流体运动的不稳定性质在这种噪音的存在，以及这种不稳定性如何表现为混沌和湍流运动在自然界中观察到。该项目将为本科生和研究生提供参与的机会，研究结果将得到广泛传播。对确定性流体模型中的正李雅普诺夫指数和许多不稳定现象进行严格的分析是一项艰巨的任务，而且大多数情况下似乎超出了当前数学分析的范围。最近，在证明与流体力学相关的随机系统的不稳定性方面取得了重大进展，包括二维随机Navier-Stokes方程的伽辽金截断和与随机流体模型相关的拉格朗日流动。本项目主要关注几个相关的主要方向：1)研究随机复Ginzburg-Landau方程Galerkin截断中Lyapunov指数的不稳定性和正性；2)研究随机剪切模态强迫二维Galerkin - Navier-Stokes平稳测量中的级联不稳定性和分岔；3)研究随机混合速度平流扩散方程中Batchelor尺度的出现。这些研究都需要从随机动力系统的光滑遍历理论中开发工具来回答有关不稳定性的问题。有几个问题涉及到具有挑战性的次椭圆性问题，这些问题可以使用最近从计算代数几何中发展出来的技术来研究。例如，项目i)和ii)都涉及研究非常具有挑战性的方程特有的准椭圆结构中的简并性，并且需要新颖的思想和技术来克服这些挑战。另一方面，项目iii)旨在通过光滑的遍历流体运动来深入了解小规模地层在平流扩散中的局限性。总的来说，这些项目旨在为研究噪声存在下的流体不稳定性和混合带来新的视角和新技术。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project concerns the study of models related to the motion of a turbulent fluid. One of the defining characteristics of a turbulent fluid is its chaotic, seemingly unpredictable behavior. A widely recognized way to describe this chaotic behavior is by showing extreme exponential sensitivity with respect to initial conditions, commonly referred to as the "butterfly effect," whereby tiny changes in the state of the fluid lead to very big changes after a short amount of time. Despite its fundamental nature, there are very few mathematical tools available for rigorously verifying exponential sensitivity in a given system, particularly in turbulent systems. This project aims to develop mathematical tools to prove this sensitivity for various models in fluid mechanics in the presence of a small amount of noise. Such noise is commonly used to model the effect of unpredictable environmental effects or small-scale fluctuations of particles. The goal of this project is to gain new insights into the unstable nature of fluid motion in the presence of such noise and how this instability manifests as chaotic and turbulent motion observed in nature. The project will provide opportunities to involve undergraduate and graduate students and the results of the research will be widely disseminated.A rigorous analysis of positive Lyapunov exponents and the numerous unstable phenomena in deterministic fluid models is a daunting task and mostly appears to be out of reach of current mathematical analysis. Recently there has been significant progress in proving instability for stochastic systems related to fluid mechanics, including Galerkin truncations of the 2d stochastic Navier-Stokes equations and the Lagrangian flow associated with stochastic fluid models. This project focuses on several primary related directions: i) a study of instability and positivity of the Lyapunov exponent in Galerkin truncations for the stochastic complex Ginzburg-Landau equations ii) a study of cascading instabilities and bifurcations in the stationary measures for stochastically shear mode forced 2d Galerkin Navier-Stokes iii) a study of the emergence of the Batchelor scale in the advection diffusion equation with random mixing velocities. Each of these investigations requires developing tools from smooth ergodic theory of random dynamical systems to answer questions about instability. Several involve challenging hypoellipticity questions that can be studied using recently developed techniques from computational algebraic geometry. For instance, both the projects i) and ii) involve studying very challenging degeneracies in the hypoelliptic structure unique to the equations and require novel ideas and techniques to overcome these challenges. Project iii) on the other hand aims to give some insight into the limitations of small-scale formation in advection diffusion by smooth ergodic fluid motion. Overall, these projects aim to bring new perspectives and new techniques into the study of fluid instability and mixing in the presence of noise.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.