Stationary Inviscid Limit to Shear Flows

TL;DR

This paper proves a density result for stationary shear flows in a channel and constructs Navier-Stokes solutions close to these flows, using coercivity estimates for the Rayleigh operator.

Contribution

It introduces a new density result for shear flows vanishing at boundaries and constructs near-identical Navier-Stokes solutions based on this property.

Findings

Established a density result for boundary-vanishing shear flows.

Constructed stationary Navier-Stokes solutions close to given shear flows.

Utilized coercivity estimates for the Rayleigh operator in the construction.

Abstract

In this note we establish a density result for certain stationary shear flows, $\mu(y)$, that vanish at the boundaries of a horizontal channel. We construct stationary solutions to 2D Navier-Stokes that are $\epsilon$-close in $L^\infty$ to the given shear flow. Our construction is based on a coercivity estimate for the Rayleigh operator, $R[v]$, which is based on a decomposition made possible by the vanishing of $\mu$ at the boundaries.