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

Juan Vicente Gutiérrez-Santacreu

doi:10.3934/dcds.2020142

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2020, Volume 40, Issue 5: 2593-2613. Doi: 10.3934/dcds.2020142

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

Juan Vicente Gutiérrez-Santacreu^,

Dpto. de Matemática Aplicada I, E. T. S. I. Informática, Universidad de Sevilla, Avda. Reina Mercedes, s/n, Sevilla, E-41012, Spain

^* Corresponding author: Juan Vicente Gutiérrez-Santacreu
^* Corresponding author: Juan Vicente Gutiérrez-Santacreu

Received: December 2018

Revised: May 2019

Published: March 2020

JVGS was partially supported by the Spanish Grant No. PGC2018-098308-B-I00 from Ministerio de Ciencias e Innovación - Agencia Estatal de Investigación with the participation of FEDER

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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, a 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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Mathematics Subject Classification: Primary: 35Q30, 35D30, 35D35, 35B44; Secondary: 35B65.

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