Two-dimensional incompressible viscous flow around a small obstacle

Details Two-dimensional incompressible viscous flow around a small obstacle Two-dimensional incompressible viscous flow around a small obstacle

2005-09-15

arxiv.org/abs/math/0509354v1

Math. Ann. v. 336 (2006), 449-489

Mathematics

Analysis of PDEs

Scientific paper

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 in [Comm P.D.E. 28 (2003) 349-379], where the effect of the small obstacle appears in the coefficients of the PDE and not only on 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.

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