Two scenarios on a potential smoothness breakdown for the three-dimensional Navier-Stokes equations

TL;DR

This paper constructs specific initial data for the 3D Navier-Stokes equations that lead to solutions becoming smooth either after a long time or shortly after starting, highlighting potential scenarios for singularity formation.

Contribution

It introduces two families of initial data demonstrating possible smoothness breakdown scenarios at prescribed times, advancing understanding of singularity formation in Navier-Stokes equations.

Findings

Solutions can become smooth after arbitrarily long or short times.

Large energy consumption before potential blow-up ensures global smoothness.

Constructs explicit initial data leading to controlled smoothness timelines.

Abstract

In this paper we construct two families of initial data being arbitrarily large under any scaling-invariant norm for which their corresponding weak solution to the three-dimensional Navier-Stokes equations become smooth on either $[0,T_1]$ or $ [T_2,\infty)$, respectively, where $T_1$ and $T_2$ are two times prescribed previously. In particular, $T_1$ can be arbitrarily large and $T_2$ can be arbitrarily small. Therefore, possible formation of singularities would occur after a very long or short evolution time, respectively. We further prove that if a large part of the kinetic energy is consumed prior to the first (possible) blow-up time, then the global-in-time smoothness of the solutions follows for the two families of initial data.