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

Juan Vicente Gutiérrez-Santacreu

doi:10.3934/dcds.2020142

Article Contents

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

This issue Previous Article Attractors and their stability with respect to rotational inertia for nonlocal extensible beam equations Next Article Characterization of minimizable Lagrangian action functionals and a dual Mather theorem

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: November 30, 2018

Revised: April 30, 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

Abstract Full Text(HTML) Related Papers Cited by

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.

Keywords:

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

Citation:

Full Text(HTML)

Related Papers

Cited by

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.
[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.
[3]	M. Cannone, Ondelettes, Paraproduits et Navier-Stokes, Diderot Editeur, Paris, 1995.
[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.
[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.
[6]	L. Escauriaza, G. 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.
[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.
[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.
[9]	I. Gallagher, D. 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.
[10]	E. Hopf, Über die aufangswertaufgabe für die hydrodynamischen Grundgliechungen, Mathematische Nachrichten, 4 (1951), 213-231. doi: 10.1002/mana.3210040121.
[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.
[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.
[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.
[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.
[15]	J. Leray, Essai sur les mouvements d'un liquide visqueux emplissant l'espace, Acta Mathematica, 63 (1934), 193-248.
[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.
[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.