Hydrodynamic Stability at High Reynolds Number and Transition Threshold Problem

Explore the intricacies of hydrodynamic stability theory and the transition threshold problem in this 46-minute lecture by Zhifei Zhang for the International Mathematical Union. Delve into the mechanisms behind laminar flow instability and turbulence transition at high Reynolds numbers. Examine key physical effects such as 3-D lift-up, inviscid damping, enhanced dissipation, and boundary layer behavior. Survey recent advancements in linear inviscid damping and enhanced dissipation for shear flows. Gain insights into the proof of transition threshold for 3-D Couette flow in a finite channel, including key ideas and main components. Follow along with topics like Reynolds experiments, Navier-Stokes equations, eigenvalue analysis, subcritical transition, and various mathematical models. Conclude with an exploration of open problems in the field, providing a comprehensive overview of this complex area of fluid dynamics.

Intro
Reynolds experiment in 1883
Mathematical model Navier-Stokes equations
Examples of laminar flow
Eigenvalue analysis
Subcritical transition
Transition threshold problem
Numerics and asymptotic analysis results
Mathematical analysis results
Key factors influencing the threshold
Linear inviscid damping: monotone flow
Linear inviscid damping: Kolmogorov flow
Linear inviscid damping: methods of the proof The key ingredient of the proof is to solve the inhomogeneous
Nonlinear inviscid damping
Linear enhanced dissipation
Chapman toy model Consider a toy model introduced by Chapman
Chapman tay model: scaling analysis
Chapman tay model: secondary instability
Chapman toy model: transition route
Perturbation NS system
Secondary instability of wall mode
Transition threshold for 3-D Couette flow
Key ingredients(l): space-time estimates
Key ingredients (ll): exclude secondary instability
Key ingredients(lll): energy functional
Open problems