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Remarks on the breakdown of smooth solutions for the 3-D Euler equations

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The authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3-D Euler equations. In other words, if a solution of the Euler equations is initially smooth and loses its regularity at some later time, then the maximum vorticity necessarily grows without bound as the critical time approaches; equivalently, if the vorticity remains bounded, a smooth solution persists.

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Communicated by L. Nirenberg

Partially supported by O.N.R. Contract No. N00014-76-C-0316 and N.S.F. Grant No. MCS-81-01639

Partially supported by N.S.F. Grant No. MCS-82-00171

Partially supported by N.S.F. Grant No. MCS-81-02360

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Beale, J.T., Kato, T. & Majda, A. Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun.Math. Phys. 94, 61–66 (1984).

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