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Existence of axisymmetric weak solutions of the 3-D Euler equations for near-vortex-sheet initial data. (English) Zbl 0903.35050
Summary: We study the initial value problem for the 3-D Euler equation when the fluid is inviscid and incompressible, and flows with axisymmetry and without swirl. On the initial vorticity \(\omega_0\), we assume that \(\omega_0/r\) belongs to \(L(\log L (\mathbb{R}^3))^{\alpha}\) with \(\alpha >1/2\), where \(r\) is the distance to an axis of symmetry. To prove the existence of weak global solutions, we prove first a new a priori estimate for the solution.

MSC:
35Q35 PDEs in connection with fluid mechanics
76B47 Vortex flows for incompressible inviscid fluids
35D05 Existence of generalized solutions of PDE (MSC2000)
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