BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Jacob Bedrossian (University of Maryland) DTSTART:20200423T150000Z DTEND:20200423T155000Z DTSTAMP:20260404T145132Z UID:IMS/6 DESCRIPTION:Title: The power spectrum of passive scalar turbulence in the Batchelor regime \nby Jacob Bedrossian (University of Maryland) as part of PDE seminar via Zoom\n\n\nAbstract\nIn 1959\, Batchelor predicted that passive scalars advected in fluids at finite Reynolds number with small diffusivity κ sh ould display a |k|−1 power spectrum over a small-scale inertial range in a statistically stationary experiment. This prediction has been experimen tally and numerically tested extensively in the physics and engineering li terature and is a core prediction of passive scalar turbulence. Together w ith Alex Blumenthal and Sam Punshon-Smith\, we have provided the first mat hematically rigorous proof of this prediction for a scalar field evolving by advection-diffusion in a fluid governed by the 2D Navier-Stokes equatio ns and 3D hyperviscous Navier-Stokes equations in a periodic box subjected to stochastic forcing at arbitrary Reynolds number. These results are pro ved by studying the Lagrangian flow map using infinite dimensional extensi ons of ideas from random dynamical systems. We prove that the Lagrangian f low has a positive Lyapunov exponent (Lagrangian chaos) and show how this can be upgraded to almost sure exponential (universal) mixing of passive s calars at zero diffusivity and further to uniform-in-diffusivity mixing. T his in turn is a sufficiently precise understanding of the low-to-high fre quency cascade to deduce Batchelor's prediction.\n LOCATION:https://stable.researchseminars.org/talk/IMS/6/ END:VEVENT END:VCALENDAR