Stability, Mixing, and Stochastics in Hydrodynamics

University Of Utah, Salt Lake City UT

Investigators

Abstract

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.