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

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 $u^{\nu }$ and a fixed (laminar) regular Euler solution $\overline{u}$ with similar initial conditions and body force. We show an asymptotic upper bound $C\left|\right|\overline{u}|{|}_{L^{\infty }}^{3}T$ 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.

KEYWORDS:

• Navier–Stokes equation
• inviscid limit
• boundary regularity
• blow-up technique
• layer separation

MSC SUBJECT CLASSIFICATION (2010):

• 76D05
• 35Q30

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 $D\overline{u}$ should be interpreted as its largest absolute eigenvalue, which corresponds to the maximum expansion/contraction rate.

Additional information

Funding

The first author was partially supported by the NSF grant: DMS 2306852. The second author was partially supported by the NSF grant: DMS 2054888.