Nonlinear artificial boundary conditions for the Navier‐Stokes equations in an aperture domain

Abstract

AbstractWe consider the Dirichlet problem for the stationary Navier‐Stokes system in a plane domain Ω, with two angular outlets to infinity. It is known that, under appropriate decay and smallness assumptions, this problem admits solutions with main asymptotic terms in Jeffrey‐Hamel form. We will approach these solutions by constructing an approximating problem in the domain ΩR, which is the intersection of Ω with a sufficiently large circle. The main difficulty, in contrast to the corresponding linear problem, arises from the fact that the main asymptotic term is not known explicitly. Here, we create nonlinear, but local, artificial boundary conditions which involve second order differential operators on the truncation arcs. Unlike for the analogous three‐dimensional exterior problem, we are able to show the existence of weak solutions to the approximating problem without smoothness nor smallness assumptions. For small data, we prove that the solutions of the approximating problem are unique and regular. Finally, we reach the main goal of this work, i.e. we obtain error estimates in weighted Hölder spaces which are asymptotically precise as R tends to infinity. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)