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

  • * Corresponding author: Juan Vicente Gutiérrez-Santacreu

    * Corresponding author: Juan Vicente Gutiérrez-Santacreu

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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  • 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.

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


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