Zhi "George" Lin, University of North Carolina at Chapel Hill

Evolution of the Probability Measure for the Majda Model: New Invariant Measures and Breathing PDFs

Friday January 26th, 4pm, Phillips 383
(refreshments served in Phillips 330 starting at 3:30)

Abstract: In 1993, A. J. Majda proposed a simple, random shear model from which scalar intermittency was rigorously predicted for the invariant probability measure of passive tracers. In this work, we present an integral formulation for the tracer measure, which leads to a new, comprehensive study on its temporal evolution based on Monte Carlo simulation and direct numerical integration. An interesting, non-monotonic "breathing" phenomena is discovered from these results and carefully defined, with a solid example for special initial data to predict such phenomena. Further, the "breathing" PDF is recovered as a new invariant measure in a distinguished time scale in the diffusionless limit. Rigorous asymptotic analysis is also performed to identify the Gaussian core of the invariant measures, and the critical rate at which the heavy, stretched exponential regime propagates towards the tail as a function of time is calculated.

This is a joint work with Roberto Camassa and Richard McLaughlin at UNC at Chapel Hill.