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Remarks on the breakdown of smooth solutions for the 3-D Euler equations. (English) Zbl 0573.76029
The authors consider the Euler equations for the motion of an incompressible, inviscid fluid in the whole space. They show that, if a solution is initially smooth and loses its regularity at some later time, then the maximum vorticity necessarily grows without bound as the critical time approaches. Therefore, it is not possible for other kinds of singularities to form before the vorticity becomes unbounded. On the other hand, if the vorticity remains bounded, a smooth solution persists. In other words, the maximum norm of the vorticity controls the breakdown of smooth solutions for the 3-D Euler equations. The same result, with the vorticity substituted by the deformation tensor, has been recently proved by G. Ponce in Commun. Math. Phys. 98, 349-353 (1985; Zbl 0589.76040).
Reviewer: P.Secchi

MSC:
76Bxx Incompressible inviscid fluids
35Q99 Partial differential equations of mathematical physics and other areas of application
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