#### Vol. 1, No. 1, 2008

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Vanishing viscosity plane parallel channel flow and related singular perturbation problems

### Anna Mazzucato and Michael Taylor

Vol. 1 (2008), No. 1, 35–93
##### Abstract

We study a special class of solutions to the three-dimensional Navier–Stokes equations ${\partial }_{t}{u}^{\nu }+{\nabla }_{{u}^{\nu }}{u}^{\nu }+\nabla {p}^{\nu }=\nu \Delta {u}^{\nu }$, with no-slip boundary condition, on a domain of the form $\Omega =\left\{\left(x,y,z\right):0\le z\le 1\right\}$, dealing with velocity fields of the form ${u}^{\nu }\left(t,x,y,z\right)=\left({v}^{\nu }\left(t,z\right),{w}^{\nu }\left(t,x,z\right),0\right)$, describing plane-parallel channel flows. We establish results on convergence ${u}^{\nu }\to {u}^{0}$ as $\nu \to 0$, where ${u}^{0}$ solves the associated Euler equations. These results go well beyond previously established ${L}^{2}$-norm convergence, and provide a much more detailed picture of the nature of this convergence. Carrying out this analysis also leads naturally to consideration of related singular perturbation problems on bounded domains.

##### Keywords
Navier–Stokes equations, viscosity, boundary layer, singular perturbation
##### Mathematical Subject Classification 2000
Primary: 35B25, 35K20, 35Q30