Vanishing Viscosity Plane Parallel Channel Flow and Related Singular Perturbation Problems

We study a special class of solutions to the 3D Navier-Stokes equations ∂tu +∇uνu +∇p = ν∆u , 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(t, x, y, z) = (v(t, z), w(t, x, z), 0), describing plane-parallel channel flows. We establish results on convergence u → u as ν → 0, where u solves the associated Euler equations. These results go well beyond previously established L-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. ∗2000 Math Subject Classification. 35Q30, 35K20, 35B25