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
DOI:
https://doi.org/10.1090/S0033-569X-07-01059-4
Published electronically:
July 9, 2007
MathSciNet review:
2354884
Full-text PDF Free Access
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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 $\varepsilon >0$. 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.
References
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References
- Benedetto, D., Cagliotti, E. and Marchioro, C., On the motion of a vortex ring with a sharply concentrated vorticity, Math. Methods Appl. Sci. 23 (2000), 147–168. MR 1738349 (2001b:76016)
- Caglioti, E. and Marchioro, C., Bounds on the growth of the velocity support for the solutions of the Vlasov-Poisson equation in a torus, J. Statist. Phys. 100 (2000), 659–677. MR 1788481 (2001i:82063)
- Constantin, P., Geometric statistics in turbulence. SIAM Rev. 36 (1994), 73–98. MR 1267050 (95d:76057)
- Iftimie, D., Lopes Filho, M. C. and Nussenzveig Lopes, H. J., Two dimensional incompressible ideal flow around a small obstacle, Commun. P. D. E. 28 (2003), 349–379. MR 1974460 (2004d:76009)
- Iftimie, D., Lopes Filho, M. C. and Nussenzveig Lopes, H. J., Large time behavior for vortex evolution in the half-plane, Comm. Math. Phys. 237 (2003), 441–469. MR 1993334 (2004d:76020)
- Iftimie, D., Lopes Filho, M. C. and Nussenzveig Lopes, H. J., On the large-time behavior of two-dimensional vortex dynamics Phys. D 179 (2003), 153–160. MR 1984383 (2004d:76021)
- Iftimie, D., Sideris, T.C. and Gamblin, P., On the evolution of compactly supported planar vorticity, Comm. Part. Diff. Eqns., 24 (1999), 1709–1730. MR 1708106 (2000d:76030)
- Kikuchi, K., Exterior problem for the two-dimensional Euler equation, J. Fac. Sci. Univ. Tokyo, Sect 1A, 30 (1983), 63–92. MR 700596 (84g:35151)
- Maffei, C. and Marchioro, C., A confinement result for axisymmetric fluids, Rend. Sem. Mat. Univ. Padova, 105 (2001), 125–137. MR 1834985 (2002c:76029)
- Marchioro, C., Bounds on the growth of the support of a vortex patch, Comm. Math. Phys. 164 (1994), 507–524. MR 1291243 (95f:76012)
- Marchioro, C., On the growth of the vorticity support for an incompressible non-viscous fluid in a two-dimensional exterior domain, Math. Meth. Appl. Sci., 19 (1996), 53–62. MR 1365263 (97i:35146)
- Marchioro, C., A confinement result on a quasi-geostrophic flow in the $f$-plane, Z. Angew. Math. Phys. 47 (1996), 16–27. MR 1408668 (97g:86009)
- Marchioro, C., On the inviscid limit for a fluid with a concentrated vorticity, Comm. Math. Phys. 196 (1998), 53–65. MR 1643505 (99f:76041)
- Marchioro, C., On the localization of the vortices, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 1 (1998), 571–584. MR 1662329 (99k:76035)
- Marchioro, C., Large smoke rings with concentrated vorticity, J. Math. Phys. 40 (1999), 869–883. MR 1674263 (99k:76036)
- Marchioro, C. and Pulvirenti, M., Vortices and localization in Euler flows, Comm. Math. Phys., 154 (1993), 49–61. MR 1220946 (94i:35172)
- Serfati, Ph., Bornes en temps des characteristiques des l’equation d’Euler 2D à tourbillon positif et le localization pour le modele point-vortex, preprint, 1998.
- Warner, F., Foundations of Differential Equations and Lie Groups, Scott, Foresman and Co., Glenview, IL, 1971. MR 0295244 (45:4312)
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Additional Information
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
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