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

TL;DR

This paper investigates potential singularities in 3D Euler and Navier-Stokes equations, extending nonexistence results and deriving new a priori estimates through generalized self-similar transforms.

Contribution

It extends previous nonexistence results for self-similar singularities and introduces new a priori estimates for 3D Euler and Navier-Stokes equations using generalized transforms.

Findings

Nonexistence of certain self-similar singularities in 3D Euler.

New a priori estimates for Euler and Navier-Stokes equations.

Implications for the blow-up problem in fluid dynamics.

Abstract

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.