Two dimensional incompressible ideal flow around a small obstacle. (English) Zbl 1094.76007

Summary: We study the asymptotic behavior of incompressible, ideal, time-dependent two dimensional flow in the exterior of a single smooth obstacle when the size of the obstacle becomes very small. Our main purpose is to identify the equation satisfied by the limit flow. We will see that the asymptotic behavior depends on \(\gamma\), the circulation around the obstacle. For smooth flow around a single obstacle, \(\gamma\) is a conserved quantity which is determined by the initial data. We will show that if \(\gamma=0\), the limit flow satisfies the standard incompressible Euler equations in the full plane but, if \(\gamma\neq 0\), the limit equation acquires an additional forcing term. We treat this problem by first constructing a sequence of approximate solutions to the incompressible 2D Euler equation in the full plane from the exact solutions obtained when solving the equation on the exterior of each obstacle and then passing to the limit on the weak formulation of the equation. We use an explicit treatment of the Green’s function of the exterior domain based on conformal maps, a priori estimates obtained by carefully examining the limiting process and the Div-Curl Lemma, together with a standard weak convergence treatment of the nonlinearity for the passage to the limit.


76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35B40 Asymptotic behavior of solutions to PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
35Q35 PDEs in connection with fluid mechanics
76B47 Vortex flows for incompressible inviscid fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: DOI arXiv


[1] Bell S, Rocky Mt J Math 17 (1) pp 23– (1987) · Zbl 0626.30005
[2] Chemin J-Y., Ann Sci Ecole Norm Sup ser 26 (4) pp 517– (1993)
[3] Delort J-M., J Amer Math Soc 4 (3) pp 553– (1991)
[4] Hounie J, Nonlinear Analysis TMA 35 (7) pp 871– (1999) · Zbl 0920.35111
[5] Iftimie D, Comm Part Diff Eqns 24 (9) pp 1709– (1999)
[6] Kikuchi K., J Fac Sci Univ Tokyo Sect 1A Math 30 (1) pp 63– (1983)
[7] Ladyzhenskaya, OA. 1963.The mathematical theory of viscous incompressible flow., Revised English edition ; xiv+184 pp.New York-London: Gordon and Breach Science Publishers. Translated from the Russian by Richard A. Silverman
[8] Lockhart R., Trans Amer Math Soc 301 (1) pp 1– (1987)
[9] Lopes Filho MC, Ann Inst H Poincaré–Anal Non Linéaire 17 (3) pp 371– (2000) · Zbl 0965.35110
[10] Marchioro C., Comm Math Phys 164 (3) pp 507– (1994) · Zbl 0839.76010
[11] Marchioro C., Math Meth Appl Sci 19 (1) pp 53– (1996) · Zbl 0845.35089
[12] Saffman, PG. 1992.Vortex Dynamicsxii+311 pp.New York: Cambridge Monographs on Mech and Appl Math Cambridge Univ Press.
[13] Temam R., Theory and numerical analysis. Studies in Math and Appl v 2 pp x+500 pp.– (1977) · Zbl 0383.35057
[14] Teman R, J Diff Eqs 179 (2) pp 647– (2002) · Zbl 0997.35042
[15] Warner, FW. 1971.Foundations of Differentiable Manifolds and Lie Groupsviii+270 pp.Glenview, IL: Scott, Foresman and Co.
[16] Yudovich VI., Zh Vych Mat 3 pp 1032– (1963)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.