Layer separation of the 3D incompressible Navier-Stokes equation in a bounded domain

Abstract

We provide an unconditional $L^2$ upper bound for the boundary layer separation of Leray-Hopf solutions in a smooth bounded domain. By layer separation, we mean the discrepancy between a (turbulent) low-viscosity Leray-Hopf solution $u^{\nu }$ and a fixed (laminar) regular Euler solution $\bar{u}$ with similar initial conditions and body force. We show an asymptotic upper bound $C\Vert \bar{u}{\Vert }_{L^{\infty }}^{3}T$ on the layer separation, anomalous dissipation, and the work done by friction. 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.

33 pages, 2 figures

References in corpus (2)

• Anomalous dissipation for the forced 3D Navier-Stokes equations
• Boundary Vorticity Estimates for Navier-Stokes and Application to the Inviscid Limit

Cited by in corpus (1)

• Vorticity interior trace estimates and higher derivative estimates via blow-up method