Formation of small scales in nonlinear PDEs

Inviscid damping and enhanced dissipation of the boundary layer for 2D Navier-Stokes linearized around Couette flow in a channel

Siming He

Duke University


In this talk, I will present a recent result concerning the 2D Navier-Stokes equations linearized around the Couette flow $(y,0)$ in the periodic channel $ [-1,1]$ with no-slip boundary conditions in the vanishing viscosity $u \rightarrow 0$ limit. We split the vorticity evolution into the free-evolution (without a boundary) and a boundary corrector that is exponentially localized to an $O(u^{1/3})$ boundary layer. If the initial vorticity perturbation is supported away from the boundary, we show inviscid damping of emph{both} the velocity emph{and} the vorticity associated with the boundary layer. We also observe that both velocity and vorticity satisfy the expected $O(\exp(-\delta u^{1/3}t))$ enhanced dissipation in addition to the inviscid damping. This is joint work with Jacob Bedrossian.