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Evolution of compactly supported vorticity. (Évolution de tourbillon à support compact.) (French) Zbl 1103.76314

Proceedings of the conference on partial differential equations, Saint-Jean-de-Monts, France, May 31–June 4, 1999. Exp. Nos. I–XIX (1999). Nantes: Université de Nantes (ISBN 2-86939-146-3/pbk). Exp. No. 4, 8 p. (1999).
Summary: (Translation from the French) We consider the incompressible Euler equation in the plane. We show that when the vorticity distribution is positive and compactly supported, its support grows no faster than \(O[(t\log t)]^{1/4}\), an improvement of the bound \(O(t^{1/3})\) obtained by C. Marchioro. We also give an example, when the vorticity distribution changes sign, of an initial vorticity distribution for which the growth of the diameter of the support is exactly \(O(t)\). Finally, in the case of the half-plane and of a positive initial vorticity distribution with compact support, we show that the center of mass is displaced in parallel to the axis at a rate bounded below by a positive constant, and moreover that the distance between a point of the support and the axis is at most \(O[(t\log t)^{1/3}]\).
{See also D. Iftimie, T. C. Sideris and P. Gamblin, Commun. Partial Differ. Equ. 24, No. 9-10, 1709–1730 (1999; Zbl 0937.35137).}
For the entire collection see [Zbl 0990.00047].

MSC:

76B47 Vortex flows for incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids

Citations:

Zbl 0937.35137
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