Abstract. Let ω be the vorticity of a stationary solution of the two-dimensional Navier-Stokes equations with a drift term parallel to the boundary in the half-plane Ω + = {(x, y) ∈ R 2 | y > 1}, with zero Dirichlet boundary conditions at y = 1 and at infinity, and with a small force term of compact support. Then |xyω(x, y)| is uniformly bounded in Ω + . The proof is given in a specially adapted functional framework, and the result is a key ingredient for obtaining information on the asymptotic behavior of the velocity at infinity. 1. Introduction. In this paper we consider the steady Navier-Stokes equations in a half-plane Ω + = {(x, y) ∈ R 2 | y > 1} with a drift term parallel to the boundary, a small driving force of compact support, with zero Dirichlet boundary conditions at the boundary of the half-plane and at infinity. See [14] and [15] for a detailed motivation of this problem. Existence of a strong solution for this system was proved in [14] together with a basic bound on the decay at infinity, and the existence of weak solutions was shown in [15]. By elliptic regularity weak solutions are smooth, and their only possible shortcoming is the behavior at infinity, since the boundary condition may not be satisfied there in a pointwise sense. In [15] it was also shown that for small forces there is only one weak solution. This unique weak solution therefore coincides with the strong solution and as a consequence satisfies the boundary condition at infinity in a pointwise sense.The aim of this paper is to provide additional information concerning the behavior of this solution at infinity by analyzing the solution obtained in [14] in a more stringent functional setting. More precisely, we obtain more information on the decay behavior of the vorticity of the flow. Bounds on vorticity as a step towards bounds on the velocity are a classical procedure in asymptotic analysis of fluid flows (see the seminal papers [7], [8], and [1]). In [14] and the current work, the equation for the vorticity is Fourier-transformed with respect to the coordinate x parallel to the wall, and then rewritten as a dynamical system with the coordinate y perpendicular to the wall playing the role of time. In this setting information on the behavior of the vorticity at infinity is studied by analyzing the Fourier transform at k = 0, with k the Fourier conjugate variable of x. In the present work, we also control the derivative of the Fourier transform of the vorticity, which yields more precise decay estimates for the vorticity and the velocity field in direct space than the ones found in [14]. Our proof

We consider the Navier-Stokes equations in a half-plane with a drift term parallel to the boundary and a small source term of compact support. We provide detailed information on the behavior of the velocity and the vorticity at infinity in terms of an asymptotic expansion at large distances from the boundary. The expansion is universal in the sense that it only depends on the source term through some multiplicative constants. This expansion is identical to the one for the problem of an exterior flow around a small body moving at constant velocity parallel to the boundary, and can be used as an artificial boundary condition on the edges of truncated domains for numerical simulations.

We discuss artificial boundary conditions for stationary Navier–Stokes flows past bodies in the half-plane, for a range of low Reynolds numbers. When truncating the half-plane to a finite domain for numerical purposes, artificial boundaries appear. We present an explicit Dirichlet condition for the velocity at these boundaries in terms of an asymptotic expansion for the solution to the problem. We show a substantial increase in accuracy of the computed values for drag and lift when compared with results for traditional boundary conditions. We also analyze the qualitative behavior of the solutions in terms of the streamlines of the flow. The new boundary conditions are universal in the sense that they depend on a given body only through one constant, which can be determined in a feed-back loop as part of the solution process

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