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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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