Title: Layer separation for the 3D Navier--Stokes equation in a bounded domain

Abstract: We provide an unconditional L2L2 upper bound for the boundary layer separation of 3D Leray--Hopf solutions in a smooth bounded domain. By layer separation, we mean the discrepancy between a (turbulent) low-viscosity Leray--Hopf solution uνuν and a fixed (laminar) regular Euler solution u¯u¯ with initial conditions close in L2L2. Layer separation appears in physical and numerical experiments near the boundary, and we bound it asymptotically by C∥u¯∥3L∞tC‖u¯‖L∞3t. This extends the previous result when the Euler solution is a regular shear in a finite channel. The key estimate is to control the boundary vorticity in a way that does not degenerate in the vanishing viscosity limit.

This is joint work with Alexis Vasseur.