Abstract
We provide an unconditional L2 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 and a fixed (laminar) regular Euler solution with similar initial conditions and body force. We show an asymptotic upper bound 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.
Acknowledgments
The authors would like to thank American Institute of Mathematics for the workshop “Criticality and stochasticity in quasilinear fluid systems”, where this project was initiated.
Notes
1 The norm of should be interpreted as its largest absolute eigenvalue, which corresponds to the maximum expansion/contraction rate.