Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Confinement of vorticity in two dimensional ideal incompressible exterior flow


Authors: D. Iftimie, M. C. Lopes Filho and H. J. Nussenzveig Lopes
Journal: Quart. Appl. Math. 65 (2007), 499-521
MSC (2000): Primary 76B47; Secondary 35Q35
Published electronically: July 9, 2007
MathSciNet review: 2354884
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Abstract: In [Math. Meth. Appl. Sci. 19 (1996) 53-62], C. Marchioro examined the problem of vorticity confinement in the exterior of a smooth bounded domain. The main result in Marchioro's paper is that solutions of the incompressible 2D Euler equations with compactly supported nonnegative initial vorticity in the exterior of a connected bounded region have vorticity support with diameter growing at most like $ \mathcal{O}(t^{(1/2)+\varepsilon})$, for any $ \varepsilon$ $ \vare>0$. In addition, if the domain is the exterior of a disk, then the vorticity support is contained in a disk of radius $ \mathcal{O}(t^{1/3})$. The purpose of the present article is to refine Marchioro's results. We will prove that, if the initial vorticity is even with respect to the origin, then the exponent for the exterior of the disk may be improved to $ 1/4$. For flows in the exterior of a smooth, connected, bounded domain we prove a confinement estimate with exponent $ 1/2$ (i.e. we remove the $ \varepsilon$) and in certain cases, depending on the harmonic part of the flow, we establish a logarithmic improvement over the exponent $ 1/2$. The main new ingredients in our approach are: (1) a detailed asymptotic description of solutions to the exterior Poisson problem near infinity, obtained by the use of Riemann mappings; (2) renormalized energy estimates and bounds on logarithmic moments of vorticity and (3) a new a priori estimate on time derivatives of logarithmic perturbations of the moment of inertia.


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D. Iftimie
Affiliation: Université de Lyon, Université Lyon 1, CNRS UMR 5208, Institut Camille Jordan, Bât. Braconnier, 43, Blvd. du 11 Novembre 1918, F-69622 Villeurbanne cedex, France
Email: dragos.iftimie@univ-lyon1.fr

M. C. Lopes Filho
Affiliation: Departamento de Matematica, IMECC-UNICAMP, Caixa Postal 6065, Campinas, SP 13083-970, Brasil
Email: mlopes@ime.unicamp.br

H. J. Nussenzveig Lopes
Affiliation: Departamento de Matematica, IMECC-UNICAMP, Caixa Postal 6065, Campinas, SP 13083-970, Brasil
Email: hlopes@ime.unicamp.br

DOI: https://doi.org/10.1090/S0033-569X-07-01059-4
Received by editor(s): September 12, 2006
Published electronically: July 9, 2007
Additional Notes: Research supported in part by CNPq grant #300.962/91-6
Research supported in part by CNPq grant #300.158/93-9 and FAEP grant #1148/99
Article copyright: © Copyright 2007 Brown University


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