Vanishing viscosity plane parallel channel flow and related singular perturbation problems. (English) Zbl 1160.35329

Mazzucato, Anna; Taylor, Michael

Anal. PDE 1, No. 1, 35-93 (2008).

Summary: 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 \(\Omega = {(x,y,z) : 0 \leq z \leq 1}\), dealing with velocity fields of the form \(u^\nu (t,x,y,z) = (v^\nu (t,z),w^\nu (t,x,z),0)\), describing plane-parallel channel flows. We establish results on convergence \(u^\nu \rightarrow u^0\) as \(\nu \rightarrow 0\), 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.

MSC:

35B25	Singular perturbations in context of PDEs
35K20	Initial-boundary value problems for second-order parabolic equations
35Q30	Navier-Stokes equations

Keywords:

Navier-Stokes equations; viscosity; boundary layer; singular perturbation