Colloquium-February 10, 2017

Friday, February 10, 2017 (3:15 p.m. in Yost 306)

Title: Subcritical transition and stability in the 3D Navier-Stokes equations at high Reynolds number

Speaker: Jacob Bedrossian (Assistant Professor, University of Maryland)

Abstract: We study small disturbances of the plane periodic Couette flow in the 3D Navier-Stokes equations in the high Reynolds number limit. The Couette flow is the simplest equilibrium in all of fluid mechanics, yet many unanswered questions still remain about its nonlinear instability. Indeed, linearized theory predicts that the flow is stable at all Reynolds numbers, however, computer simulations and related physical experiments all suggest nonlinear instability at sufficiently high Reynolds numbers. This phenomenon is known in fluid mechanics as subcritical transition. In our work, we are interested in expanding our mathematically rigorous understanding of this effect. For sufficiently smooth perturbations we sharply estimate the size of the nonlinear basin of stability as a scaling law in Reynolds number and prove that all possible instabilities near the threshold are driven only by one specific kind of instability. For rougher data we estimate we given an estimate of the nonlinear basin of stability which agrees with existing numerical experiments. In all these regimes, the fast mixing of the solution due to the mean shear dominates the dynamics for long times, driving a rapid homogenization via mixing-enhanced viscous dissipation and inviscid damping —  hydrodynamic analogue of Landau damping in plasma physics.  Joint work with Pierre Germain and Nader Masmoudi.

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