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

We study a special class of solutions to the three-dimensional Navier–Stokes equations tuν + uνuν + pν = νΔuν, with no-slip boundary condition, on a domain of the form Ω = {(x,y,z) : 0 z 1}, dealing with velocity fields of the form uν(t,x,y,z) = (vν(t,z),wν(t,x,z),0), describing plane-parallel channel flows. We establish results on convergence uν u0 as ν 0, where u0 solves the associated Euler equations. These results go well beyond previously established L2-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.

Navier–Stokes equations, viscosity, boundary layer, singular perturbation
Mathematical Subject Classification 2000
Primary: 35B25, 35K20, 35Q30
Received: 17 December 2007
Revised: 19 March 2008
Accepted: 30 May 2008
Published: 2 October 2008
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