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