On the generalized self-similar singularities for the Euler and the Navier–Stokes equations

In this paper we study blow-up rates and the blow-up profiles of possible asymptotically self-similar singularities of the Euler and the Navier–Stokes equations, where the sense of convergence and self-similarity are considered in various generalized senses. We improve substantially, in particular, the previous nonexistence results of self-similar/asymptotically self-similar singularities. Generalization of the self-similar transforms is also considered, and by appropriate choice of the parameterized transform we obtain new a priori estimates for the Euler and the Navier–Stokes equations depending on a free parameter. © 2010 Elsevier Inc. All rights reserved.