Remarks on the breakdown of smooth solutions for the 3-D Euler equations
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 \textit{G. Ponce} in Commun. Math. Phys. 98, 349-353 (1985; Zbl 0589.76040).
incompressible, inviscid fluid in the whole space
scientific article; zbMATH DE number 3915851
scientific article