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Incompressible Flow Around a Small Obstacle and the Vanishing Viscosity Limit

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Abstract

In this article we consider viscous flow in the exterior of an obstacle satisfying the standard no-slip boundary condition at the surface of the obstacle. We seek conditions under which solutions of the Navier-Stokes system in the exterior domain converge to solutions of the Euler system in the full space when both viscosity and the size of the obstacle vanish. We prove that this convergence is true assuming two hypotheses: first, that the initial exterior domain velocity converges strongly in L 2 to the full-space initial velocity and second, that the diameter of the obstacle is smaller than a suitable constant times viscosity, or, in other words, that the obstacle is sufficiently small. The convergence holds as long as the solution to the limit problem is known to exist and stays sufficiently smooth. This work complements the study of incompressible flow around small obstacles, which has been carried out in [4–6].

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Authors and Affiliations

  1. Université de Lyon, Université Lyon 1, CNRS, UMR 5208 Institut Camille Jordan, Bâtiment du Doyen Jean Braconnier, 43, Blvd du 11 Novembre 1918, F–69622, Villeurbanne Cedex, France

    Dragoş Iftimie

  2. Departamento de Matemática, IMECC, Universidade Estadual de Campinas - UNICAMP, Caixa Postal 6065, Campinas, SP, 13083-970, Brazil

    Milton C. Lopes Filho & Helena J. Nussenzveig Lopes

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  1. Dragoş Iftimie
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  2. Milton C. Lopes Filho
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  3. Helena J. Nussenzveig Lopes
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Correspondence to Helena J. Nussenzveig Lopes.

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Communicated by P. Constantin

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Iftimie, D., Lopes Filho, M.C. & Nussenzveig Lopes, H.J. Incompressible Flow Around a Small Obstacle and the Vanishing Viscosity Limit. Commun. Math. Phys. 287, 99–115 (2009). https://doi.org/10.1007/s00220-008-0621-3

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  • DOI: https://doi.org/10.1007/s00220-008-0621-3

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