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

Juan Vicente Gutiérrez-Santacreu

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arXiv · 1508.04161

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

Juan Vicente Gutiérrez-Santacreu

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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.

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Juan Vicente Gutiérrez-Santacreu. 2015-08-17. Two scenarios on a potential smoothness breakdown for the three-dimensional Navier-Stokes equations. https://arxiv.org/abs/1508.04161

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