Abstract

We study a special class of solutions to the three-dimensional Navier–Stokes equations ∂_tu^ν + ∇_u^νu^ν + ∇p^ν = νΔu^ν, 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 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 → 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.

Keywords

Navier–Stokes equations, viscosity, boundary layer, singular perturbation

Mathematical Subject Classification

Primary: 35B25, 35K20, 35Q30

Authors