Mathematische Annalen

, Volume 336, Issue 2, pp 449-489

First online:

Two-dimensional incompressible viscous flow around a small obstacle

  • D. IftimieAffiliated withInstitut Camille Jordan, Université Claude Bernard Lyon 1 Email author 
  • , M. C. Lopes FilhoAffiliated withDepartamento de Matematica, IMECC-UNICAMP
  • , H. J. Nussenzveig LopesAffiliated withDepartamento de Matematica, IMECC-UNICAMP

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In this work we study the asymptotic behavior of viscous incompressible 2D flow in the exterior of a small material obstacle. We fix the initial vorticity ω0 and the circulation γ of the initial flow around the obstacle. We prove that, if γ is sufficiently small, the limit flow satisfies the full-plane Navier–Stokes system, with initial vorticity ω0  +  γδ, where δ is the standard Dirac measure. The result should be contrasted with the corresponding inviscid result obtained by the authors in Iftimie et al. (Comm. Part. Differ. Eqn. 28, 349–379 (2003)), where the effect of the small obstacle appears in the coefficients of the PDE and not only in the initial data. The main ingredients of the proof are L p  − L q estimates for the Stokes operator in an exterior domain, a priori estimates inspired on Kato’s fixed point method, energy estimates, renormalization and interpolation.