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Iftimie, D.; Lopes Filho, M. C.; Nussenzveig Lopes, H. J.

Two-dimensional incompressible viscous flow around a small obstacle. (English) Zbl 1169.76016

Math. Ann. 336, No. 2, 449-489 (2006).
Summary: 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 \(\omega_{0}\) and the circulation \(\gamma\) of the initial flow around the obstacle. We prove that, if \(\gamma\) is sufficiently small, the limit flow satisfies the full-plane Navier-Stokes system, with initial vorticity \(\omega_{0} + \gamma\delta\), where \(\delta\) is the standard Dirac measure. The result should be contrasted with the corresponding inviscid result obtained by the authors [Commun. Partial Differ. Equations 28, No. 1–2, 349–379 (2003; Zbl 1094.76007)], 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.

Cited in 2 Reviews
Cited in 20 Documents

MSC:

76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35Q35 PDEs in connection with fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids

Keywords:

asymptotic behavior of viscous incompressible 2D flow; Navier-Stokes system

Citations:

Zbl 1094.76007
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\textit{D. Iftimie} et al., Math. Ann. 336, No. 2, 449--489 (2006; Zbl 1169.76016)
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References:

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