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 ${\partial }_t{u}^{\nu }+{\nabla }_{u^{\nu }}{u}^{\nu }+\nabla p^{\nu }=\nu \Delta u^{\nu }$, with no-slip boundary condition, on a domain of the form $\Omega =\left\{\left(x,y,z\right):0\le z\le 1\right\}$, dealing with velocity fields of the form $u^{\nu }\left(t,x,y,z\right)=\left(v^{\nu }\left(t,z\right),w^{\nu }\left(t,x,z\right),0\right)$, describing plane-parallel channel flows. We establish results on convergence $u^{\nu }\to u^0$ as $\nu \to 0$, 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.