Vanishing viscosity plane parallel channel flow and related singular perturbation problems

Mazzucato, Anna; Taylor, Michael

doi:10.2140/apde.2008.1.35

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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

DOI: 10.2140/apde.2008.1.35

Abstract

We study a special class of solutions to the three-dimensional Navier–Stokes equations $\partial_{t} u^{ν} + \nabla_{u^{ν}} u^{ν} + \nabla p^{ν} = ν Δ u^{ν}$ , with no-slip boundary condition, on a domain of the form $Ω = {(x, y, z) : 0 \leq z \leq 1}$ , dealing with velocity fields of the form $u^{ν} (t, x, y, z) = (v^{ν} (t, z), w^{ν} (t, x, z), 0)$ , describing plane-parallel channel flows. We establish results on convergence $u^{ν} \to u^{0}$ as $ν \to 0$ , 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

Milestones

Received: 17 December 2007

Revised: 19 March 2008

Accepted: 30 May 2008

Published: 2 October 2008

Authors

Anna Mazzucato
Department of Mathematics Penn State University McAllister Building University Park, PA 16802 United States

Michael Taylor
Department of Mathematics University of North Carolina CB #3250, Phillips Hall Chapel Hill, NC 27599 United States

Vol. 1, No. 1, 2008