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May  2020, 40(5): 2593-2613. doi: 10.3934/dcds.2020142

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

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

Received  December 2018 Revised  May 2019 Published  March 2020

Fund Project: 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

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.

Citation: Juan Vicente Gutiérrez-Santacreu. Two scenarios on a potential smoothness breakdown for the three-dimensional Navier–Stokes equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2593-2613. doi: 10.3934/dcds.2020142
References:
[1]

H. Beirão da Veiga, Existence and asymptotic behavior for strong solutions of the Navier-Stokes equations in the whole space, Indiana Univ. Math. J., 36 (1987), 149-166.  doi: 10.1512/iumj.1987.36.36008.  Google Scholar

[2]

J. Bourgain and N. Pavlović, Ill-posedness of the Navier-Stokes equations in a critical space in 3D, J. Funct. Anal., 255 (2008), 2233-2247.  doi: 10.1016/j.jfa.2008.07.008.  Google Scholar

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M. Cannone, Ondelettes, Paraproduits et Navier-Stokes, Diderot Editeur, Paris, 1995.  Google Scholar

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J.-Y. Chemin and I. Gallagher, Wellposedness and stability results for the Navier-Stokes equations in ${ \mathbb R}^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 599-624.  doi: 10.1016/j.anihpc.2007.05.008.  Google Scholar

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J.-Y. Chemin and I. Gallagher, Large, global solutions to the Navier-Stokes equations, slowly varying in one direction, Trans. Amer. Math. Soc., 362 (2010), 2859-2873.  doi: 10.1090/S0002-9947-10-04744-6.  Google Scholar

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L. EscauriazaG. A. Serëgin and V. Šverák, $L_{3, \infty}$-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk, 58 (2003), 3-44.  doi: 10.1070/RM2003v058n02ABEH000609.  Google Scholar

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H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Archive for Rational Mechanics and Analysis, 16 (1964), 269-315.  doi: 10.1007/BF00276188.  Google Scholar

[8]

I. Gallagher, Profile decomposition for solutions of the Navier-Stokes equations, Bull. Soc. Math. France, 129 (2001), 285-316.  doi: 10.24033/bsmf.2398.  Google Scholar

[9]

I. GallagherD. Iftimie and F. Planchon, Asymptotics and stability for global solutions to the Navier-Stokes equations, Ann. Inst. Fourier (Grenoble), 53 (2003), 1387-1424.  doi: 10.5802/aif.1983.  Google Scholar

[10]

E. Hopf, Über die aufangswertaufgabe für die hydrodynamischen Grundgliechungen, Mathematische Nachrichten, 4 (1951), 213-231.  doi: 10.1002/mana.3210040121.  Google Scholar

[11]

T. Kato, Strong $L^p$-solutions of the Navier–Stokes equation in $ \mathbb{R}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.  doi: 10.1007/BF01174182.  Google Scholar

[12]

H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.  doi: 10.1006/aima.2000.1937.  Google Scholar

[13]

Z. Lei and F. H. Lin, Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 1297-1304.  doi: 10.1002/cpa.20361.  Google Scholar

[14]

P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, 431. Chapman & Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674.  Google Scholar

[15]

J. Leray, Essai sur les mouvements d'un liquide visqueux emplissant l'espace, Acta Mathematica, 63 (1934), 193-248.   Google Scholar

[16]

A. S. Makhalov and V. P. Nikolaenko, Global solvability of three-dimensional Navier-Stokes equations with uniformly high initial vorticity, Uspekhi Mat. Nauk, 58 (2003), 79-110.  doi: 10.1070/RM2003v058n02ABEH000611.  Google Scholar

[17]

J. C. Robinson and W. Sadowski, A local smoothness criterion for solutions of the 3D Navier-Stokes equations, Rend. Semin. Mat. Univ. Padova, 131 (2014), 159-178.  doi: 10.4171/RSMUP/131-9.  Google Scholar

show all references

References:
[1]

H. Beirão da Veiga, Existence and asymptotic behavior for strong solutions of the Navier-Stokes equations in the whole space, Indiana Univ. Math. J., 36 (1987), 149-166.  doi: 10.1512/iumj.1987.36.36008.  Google Scholar

[2]

J. Bourgain and N. Pavlović, Ill-posedness of the Navier-Stokes equations in a critical space in 3D, J. Funct. Anal., 255 (2008), 2233-2247.  doi: 10.1016/j.jfa.2008.07.008.  Google Scholar

[3]

M. Cannone, Ondelettes, Paraproduits et Navier-Stokes, Diderot Editeur, Paris, 1995.  Google Scholar

[4]

J.-Y. Chemin and I. Gallagher, Wellposedness and stability results for the Navier-Stokes equations in ${ \mathbb R}^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 599-624.  doi: 10.1016/j.anihpc.2007.05.008.  Google Scholar

[5]

J.-Y. Chemin and I. Gallagher, Large, global solutions to the Navier-Stokes equations, slowly varying in one direction, Trans. Amer. Math. Soc., 362 (2010), 2859-2873.  doi: 10.1090/S0002-9947-10-04744-6.  Google Scholar

[6]

L. EscauriazaG. A. Serëgin and V. Šverák, $L_{3, \infty}$-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk, 58 (2003), 3-44.  doi: 10.1070/RM2003v058n02ABEH000609.  Google Scholar

[7]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Archive for Rational Mechanics and Analysis, 16 (1964), 269-315.  doi: 10.1007/BF00276188.  Google Scholar

[8]

I. Gallagher, Profile decomposition for solutions of the Navier-Stokes equations, Bull. Soc. Math. France, 129 (2001), 285-316.  doi: 10.24033/bsmf.2398.  Google Scholar

[9]

I. GallagherD. Iftimie and F. Planchon, Asymptotics and stability for global solutions to the Navier-Stokes equations, Ann. Inst. Fourier (Grenoble), 53 (2003), 1387-1424.  doi: 10.5802/aif.1983.  Google Scholar

[10]

E. Hopf, Über die aufangswertaufgabe für die hydrodynamischen Grundgliechungen, Mathematische Nachrichten, 4 (1951), 213-231.  doi: 10.1002/mana.3210040121.  Google Scholar

[11]

T. Kato, Strong $L^p$-solutions of the Navier–Stokes equation in $ \mathbb{R}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.  doi: 10.1007/BF01174182.  Google Scholar

[12]

H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.  doi: 10.1006/aima.2000.1937.  Google Scholar

[13]

Z. Lei and F. H. Lin, Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 1297-1304.  doi: 10.1002/cpa.20361.  Google Scholar

[14]

P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, 431. Chapman & Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674.  Google Scholar

[15]

J. Leray, Essai sur les mouvements d'un liquide visqueux emplissant l'espace, Acta Mathematica, 63 (1934), 193-248.   Google Scholar

[16]

A. S. Makhalov and V. P. Nikolaenko, Global solvability of three-dimensional Navier-Stokes equations with uniformly high initial vorticity, Uspekhi Mat. Nauk, 58 (2003), 79-110.  doi: 10.1070/RM2003v058n02ABEH000611.  Google Scholar

[17]

J. C. Robinson and W. Sadowski, A local smoothness criterion for solutions of the 3D Navier-Stokes equations, Rend. Semin. Mat. Univ. Padova, 131 (2014), 159-178.  doi: 10.4171/RSMUP/131-9.  Google Scholar

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