Long time asymptotics for two dimensional exterior flows with small circulation at infinity

Niveau: Supérieur, Licence, Bac+2
Long-time asymptotics for two-dimensional exterior flows with small circulation at infinity Thierry Gallay Institut Fourier UMR CNRS 5582 Universite de Grenoble I BP 74 38402 Saint-Martin-d'Heres, France Yasunori Maekawa Department of Mathematics Graduate School of Science Kobe University 1-1 Rokkodai, Nada-ku Kobe 657-8501, Japan February 17, 2012 Abstract We consider the incompressible Navier-Stokes equations in a two-dimensional exterior domain ?, with no-slip boundary conditions. Our initial data are of the form u0 = ??0 + v0, where ?0 is the Oseen vortex with unit circulation at infinity and v0 is a solenoidal perturbation belonging to L2(?)2 ?Lq(?)2 for some q ? (1, 2). If ? ? R is sufficiently small, we show that the solution behaves asymptotically in time like the self-similar Oseen vortex with circulation ?. This is a global stability result, in the sense that the perturbation v0 can be arbitrarily large, and our smallness assumption on the circulation ? is independent of the domain ?. 1 Introduction Let ? ? R2 be a smooth exterior domain, namely an unbounded connected open subset of the Euclidean plane with a smooth compact boundary ∂?.

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