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ON STABILITY OF PHYSICALLY REASONABLE SOLUTIONS TO THE TWO-DIMENSIONAL NAVIER–STOKES EQUATIONS

Part of: Incompressible viscous fluids Equations of mathematical physics and other areas of application Partial differential equations

Published online by Cambridge University Press:  16 May 2019

Yasunori Maekawa
Yasunori Maekawa*
Affiliation:
Department of Mathematics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto606-8502, Japan (maekawa@math.kyoto-u.ac.jp)
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Abstract

The flow past an obstacle is a fundamental object in fluid mechanics. In 1967 Finn and Smith proved the unique existence of stationary solutions, called the physically reasonable solutions, to the Navier–Stokes equations in a two-dimensional exterior domain modeling this type of flows when the Reynolds number is sufficiently small. The asymptotic behavior of their solution at spatial infinity has been studied in detail and well understood by now, while its stability has remained open due to the difficulty specific to the two-dimensionality. In this paper, we prove that the physically reasonable solutions constructed by Finn and Smith are asymptotically stable with respect to small and well-localized initial perturbations.

Keywords

Navier–Stokes equationstwo-dimensional exterior flowsflows past an obstacleOseen equations

MSC classification

Primary: 76D05: Navier-Stokes equations 76D07: Stokes and related (Oseen, etc.) flows
Secondary: 35B35: Stability 35Q30: Navier-Stokes equations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 2 , March 2021 , pp. 517 - 568
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Copyright
© Cambridge University Press 2019

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