A New Approach to Study Strong Advection Problems

Explore a new weak convergence tool for handling multiple scales in advection-diffusion models used in turbulent diffusion theories. Delve into the strategy of recasting advection-diffusion equations in moving coordinates dictated by mean advective field flow. Learn about the introduction of a fast time variable and the concept of "convergence along mean flows." Examine the sufficient structural condition on the Jacobian matrix associated with mean advective field flow for homogenization as microscopic lengthscale vanishes. Discover an example demonstrating how failure of this structural assumption leads to degenerate limit behavior. Investigate the growth in the Jacobian matrix (Lagrangian stretching) and its impact on vanishing microscopic lengthscale limit. Explore a new multiple scales convergence in weighted Lebesgue spaces and its connection to Freidlin-Wentzell theory. The 48-minute lecture, presented by Harsha Hutridurga at the Hausdorff Center for Mathematics, covers topics including mathematical setting, weak formulation, results, strategy, assumptions, change of variables, spectrum analysis, examples of periodic and almost periodic functions, multiscale convergence, constant drift, Euclidean motions, and rotation matrix.

Introduction
Mathematical Setting
Weak formulation
Results
Strategy
Assumptions
Change of variables
Spectrum of a binocular zebra
Examples of periodic functions
Examples of almost periodic functions
The strategy
Multiscale convergence
Proof
Constant drift
Euclidean motions
Rotation matrix
New notion of convergence
Summary
Examples
Papanikolaou