From Correlated to White Transport Noise in Fluid Models

Explore a mathematical conference talk examining variational frameworks for modeling stochastic dynamics in incompressible fluids, with particular focus on the interaction between large-scale fluid behavior and small-scale stochastic processes. Delve into the development of coupled equation systems that capture both scales through a variational principle formulated with Lagrangians defined on the full flow and incorporating stochastic transport constraints. Learn how the approach smooths noise terms along time, leading to stochastic dynamics as a regularization parameter approaches zero, and understand the challenges that arise when fixed noise terms result in a generalized stochastic Euler equation that becomes problematic as the regularization parameter diminishes. Examine connections with existing stochastic frameworks and discover a new variational principle that couples noise dynamics with large-scale fluid motion. Understand how this comprehensive framework provides a stochastic representation of large-scale dynamics while accounting for fine-scale components, with the main result demonstrating that the evolution of the small-scale velocity component is governed by a linearized Euler equation with random coefficients, influenced by large-scale transport, stretching, and pressure forcing. This 47-minute presentation was recorded during the thematic meeting "New trends of stochastic nonlinear systems: well-posedness, dynamics and numerics" at the Centre International de Rencontres Mathématiques in Marseille, France.