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

本计画系关于紊流流体运动模式之研究。湍流流体的一个定义特征是其混乱的，看似不可预测的行为。描述这种混沌行为的一种广泛认可的方法是通过显示相对于初始条件的极端指数敏感性，通常被称为“蝴蝶效应”，即流体状态的微小变化会在短时间内导致非常大的变化。尽管它的基本性质，有很少的数学工具可用于严格验证指数灵敏度在一个给定的系统，特别是在湍流系统。该项目旨在开发数学工具，以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灵敏度。这种噪声通常用于模拟不可预测的环境效应或颗粒的小尺度波动的影响。该项目的目标是获得新的见解，在这种噪声的存在下，流体运动的不稳定性，以及这种不稳定性如何表现为在自然界中观察到的混沌和湍流运动。该项目将为本科生和研究生提供参与的机会，研究结果将被广泛传播。对确定性流体模型中的正李雅普诺夫指数和众多不稳定现象进行严格分析是一项艰巨的任务，而且大多数情况下似乎是当前数学分析所无法达到的。近年来，与流体力学相关的随机系统不稳定性的证明取得了重大进展，包括二维随机Navier-Stokes方程的Galerkin截断和与随机流体模型相关的拉格朗日流。该项目侧重于几个主要相关方向：i）研究随机复Ginzburg-Landau方程Galerkin截断的不稳定性和正的Lyapunov指数ii）研究随机剪切模强迫2dGalerkin Navier-Stokes方程定常测度的级联不稳定性和分叉iii）具有随机混合速度的平流扩散方程中Batchelor尺度出现的研究。这些调查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。其中一些涉及具有挑战性的hypoellipticity问题，可以使用最近开发的计算代数几何技术进行研究。例如，项目i）和ii）都涉及研究方程特有的亚椭圆结构中非常具有挑战性的简并性，需要新的想法和技术来克服这些挑战。另一方面，项目iii）旨在通过平滑遍历流体运动对平流扩散中的小尺度形成的限制进行一些深入了解。总的来说，这些项目旨在为噪声环境下的流体不稳定性和混合研究带来新的视角和新的技术。该奖项反映了NSF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.