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

Tulane University, New Orleans LA

Investigators

Abstract

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.