Uniform linear inviscid damping near monotonic shear flows in the whole space

PDE Seminar
Tuesday, November 8, 2022 - 3:00pm for 1 hour (actually 50 minutes)
Skiles 006
Hao Jia – University of Minnesota – jia@umn.eduhttps://www.hao-jia-umn-math.com/
Gong Chen

In recent years tremendous progress was made in understanding the ``inviscid damping" phenomenon near shear flows and vortices, which are steady states for the 2d incompressible Euler equation, especially at the linearized level. However, in real fluids viscosity plays an important role. It is natural to ask if incorporating the small but crucial viscosity term (and thus considering the Navier Stokes equation in a high Reynolds number regime instead of Euler equations) could change the dynamics in any dramatic way. It turns out that for the perturbative regime near a spectrally stable monotonic shear flows in an infinite periodic channel (to avoid boundary layers and long wave instabilities), we can prove uniform-in-viscosity inviscid damping. The proof introduces techniques that provide a unified treatment of the classical Orr-Sommerfeld equation in a way analogous to Rayleigh equations.