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Journal of Mathematical Fluid Mechanics

, Volume 20, Issue 3, pp 937–967 | Cite as

Stability of Stationary Viscous Incompressible Flow Around a Rigid Body Performing a Translation

  • Paul Deuring
Article
  • 43 Downloads

Abstract

Suppose a rigid body moves steadily and without rotation in a viscous incompressible fluid, and the flow around the body is steady, too. Such a flow is usually described by the stationary Navier–Stokes system with Oseen term, in an exterior domain. An Oseen term arises because the velocity field is scaled in such a way that it vanishes at infinity. In the work at hand, such a velocity field, denoted by U, is considered as given. We study a solution of the incompressible evolutionary Navier–Stokes system with the same right-hand side and the same Dirichlet boundary conditions as the stationary problem, and with \(U+u_0\) as initial data, where \(u_0\) is a \(H^1\)-function. Under the assumption that the \(H^1\)-norm of \(u_0\) is small (\(u_0\) a “perturbation of U”) and that the eigenvalues of a certain linear operator have negative real part, we show that \(\Vert \nabla (v(t)-U)\Vert _2\rightarrow 0\; (t\rightarrow \infty )\) (“stability of v”), where v denotes the velocity part of the solution to the initial-boundary value problem under consideration.

Keywords

Stability Incompressible Navier–Stokes system Oseen term 

Mathematics Subject Classification

35Q30 65N30 76D05 

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© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques Pures et Appliquées Joseph LiouvilleUniversité du Littoral Côte d’OpaleCalaisFrance

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