Existence of stationary solutions of the Navier–Stokes equations in the presence of a wall

Guo, Zhengguang; Wittwer, Peter; Zhou, Yong

doi:10.1007/s00033-013-0304-6

Cited by 3 publications

(12 citation statements)

References 20 publications

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“…As in [10] we use throughout this paper a hat to indicate functions in Fourier space, i.e., we express functions f by:…”

Section: Preliminariesmentioning

confidence: 99%

“…Remark 3 Existence of solutions has been proved in [10] where we showed that ju i (x; y; z)j C=z 2 . Theorem 2 provides an explicit description of the dominant behavior of the velocity …eld and Theorem 1 implies that the bound in [10] is sharp for u 2 and u 3 , but not for u 1 :…”

Section: Introductionmentioning

confidence: 99%

“…The present paper is concerned with the asymptotic behavior of solutions, this asymptotic description is important in view of the formulation of arti…cial boundary conditions. The techniques used here take as the starting point existence results which have been obtained in [10]. Our main result is a detailed description of the asymptotic behavior of the velocity components.…”

Section: Introductionmentioning

confidence: 99%

“…The following theorems are our main results. They provide the leading order asymptotic behavior of solutions whose existence has been shown in [10]. We use the notation and de…nitions of [10] throughout this paper.…”

Section: Introductionmentioning

confidence: 99%

“…The proofs of the above theorems will be given by extracting the leading asymptotic terms from each component of the velocity in the following sections. For the convenience of the reader we recollect the main result and some expressions which have been proved in [10] and which will be used again in this paper.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Leading Order Asymptotics of Stationary Navier–stokes Flows in the Presence of a Wall

Guo

Wittwer

Zhou

2012

Math. Models Methods Appl. Sci.

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“…As in [10] we use throughout this paper a hat to indicate functions in Fourier space, i.e., we express functions f by:…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Leading Order Asymptotics of Stationary Navier–stokes Flows in the Presence of a Wall

Guo

Wittwer

Zhou

2012

Math. Models Methods Appl. Sci.

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Asymptotic behavior of D-solutions to the steady Navier–Stokes flow in an exterior domain of a half-space

Guo

Wittwer

Zhou

2019

Z. Angew. Math. Phys.

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Decay Estimates for Steady Solutions of the Navier--Stokes Equations in Two Dimensions in the Presence of a Wall

Boeckle¹,

Wittwer²

2012

SIAM J. Math. Anal.

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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

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Existence of stationary solutions of the Navier–Stokes equations in the presence of a wall

Cited by 3 publications

References 20 publications

Leading Order Asymptotics of Stationary Navier–stokes Flows in the Presence of a Wall

Leading Order Asymptotics of Stationary Navier–stokes Flows in the Presence of a Wall

Asymptotic behavior of D-solutions to the steady Navier–Stokes flow in an exterior domain of a half-space

Decay Estimates for Steady Solutions of the Navier--Stokes Equations in Two Dimensions in the Presence of a Wall

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