Electron. J. Diff. Eqns., Vol. 1998(1998), No. 26, pp. 1-17.

Existence of axisymmetric weak solutions of the 3-D Euler equations for near-vortex-sheet initial data

Dongho Chae & Oleg Yu Imanuvilov

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 assumed that $\omega_0/r$ belongs to $L(\log L (\Bbb R^3))^{\alpha}$ with $\alpha$ greater than 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.

Submitted October 9, 1998. Published October 15, 1998.
Math Subject Classification: 35Q35, 76C05.
Key Words: Euler equations, axisymmetry, weak solution.

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Dongho Chae
Department of Mathematics, Seoul National University
Seoul 151-742, Korea
e-mail: dhchae@math.snu.ac.kr
Oleg Yu Imanuvilov
Korean Institute for Advanced Study
207-43 Chungryangri-dong Dongdaemoon-ku, Seoul, Korea
e-mail: oleg@kias.kaist.ac.kr

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