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Incompressible flow around a small obstacle and the vanishing viscosity limit. (English) Zbl 1173.35628

Summary: 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 [D. Iftimie and J. Kelliher, Remarks on the vanishing obstacle limit for a 3D viscous incompressible fluid. Preprint available at http://math.umv-lyon1.fr/~iftimie/ARTICLES/viscoushrink3d.pdf, to appear in Proc. Am. Math. Soc.; D. Iftimie, M. C. Lopes Filho and H. J. Nussenzveig Lopes, Commun. Partial Differ. Equations 28, No. 1–2, 349–379 (2003; Zbl 1094.76007); Math. Ann. 336, No. 2, 449–489 (2006; Zbl 1169.76016)].

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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