Vanishing viscosity plane parallel channel flow and related singular perturbation problems

Abstract

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.