On the blow-up problem and new a priori estimates for the 3D Euler and the Navier-Stokes equations

Details On the blow-up problem and new a priori estimates for the 3D Euler and the Navier-Stokes equations On the blow-up problem and new a priori estimates for the 3D Euler and the Navier-Stokes equations

2007-11-07

arxiv.org/abs/0711.1113v3

Mathematics

Analysis of PDEs

22 pages

Scientific paper

We study blow-up rates and the blow-up profiles of possible asymptotically self-similar singularities of the 3D Euler equations, where the sense of convergence and self-similarity are considered in various sense. We extend much further, in particular, the previous nonexistence results of self-similar/asymptotically self-similar singularities obtained in \cite{cha1,cha2}. Some implications the notions for the 3D Navier-Stokes equations are also deduced. Generalization of the self-similar transforms is also considered, and by appropriate choice of the transform we obtain new \textit{a priori} estimates for the 3D Euler and the Navier-Stokes equations.

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