-
Previous Article
Characterization of minimizable Lagrangian action functionals and a dual Mather theorem
- DCDS Home
- This Issue
-
Next Article
Attractors and their stability with respect to rotational inertia for nonlocal extensible beam equations
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 |
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.
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. |
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. |
| [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. |
| [1] |
Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure and Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747 |
| [2] |
Jochen Merker. Strong solutions of doubly nonlinear Navier-Stokes equations. Conference Publications, 2011, 2011 (Special) : 1052-1060. doi: 10.3934/proc.2011.2011.1052 |
| [3] |
Peter E. Kloeden, José Valero. The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 161-179. doi: 10.3934/dcds.2010.28.161 |
| [4] |
Yukang Chen, Changhua Wei. Partial regularity of solutions to the fractional Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5309-5322. doi: 10.3934/dcds.2016033 |
| [5] |
José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091 |
| [6] |
Reinhard Farwig, Paul Felix Riechwald. Regularity criteria for weak solutions of the Navier-Stokes system in general unbounded domains. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 157-172. doi: 10.3934/dcdss.2016.9.157 |
| [7] |
Hamid Bellout, Jiří Neustupa, Patrick Penel. On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1353-1373. doi: 10.3934/dcds.2010.27.1353 |
| [8] |
Wendong Wang, Liqun Zhang. The $C^{\alpha}$ regularity of weak solutions of ultraparabolic equations. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1261-1275. doi: 10.3934/dcds.2011.29.1261 |
| [9] |
Michele Campiti, Giovanni P. Galdi, Matthias Hieber. Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613 |
| [10] |
Tian Ma, Shouhong Wang. Asymptotic structure for solutions of the Navier--Stokes equations. Discrete and Continuous Dynamical Systems, 2004, 11 (1) : 189-204. doi: 10.3934/dcds.2004.11.189 |
| [11] |
Lukáš Poul. Existence of weak solutions to the Navier-Stokes-Fourier system on Lipschitz domains. Conference Publications, 2007, 2007 (Special) : 834-843. doi: 10.3934/proc.2007.2007.834 |
| [12] |
Peter Constantin, Gregory Seregin. Global regularity of solutions of coupled Navier-Stokes equations and nonlinear Fokker Planck equations. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1185-1196. doi: 10.3934/dcds.2010.26.1185 |
| [13] |
Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure and Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35 |
| [14] |
Daniel Pardo, José Valero, Ángel Giménez. Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3569-3590. doi: 10.3934/dcdsb.2018279 |
| [15] |
Fang Li, Bo You, Yao Xu. Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4267-4284. doi: 10.3934/dcdsb.2018137 |
| [16] |
Jian-Guo Liu, Zhaoyun Zhang. Existence of global weak solutions of $ p $-Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 469-486. doi: 10.3934/dcdsb.2021051 |
| [17] |
Xiaoyu Chen, Jijie Zhao, Qian Zhang. Global existence of weak solutions for the 3D axisymmetric chemotaxis-Navier-Stokes equations with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022062 |
| [18] |
Zoran Grujić. Regularity of forward-in-time self-similar solutions to the 3D Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 837-843. doi: 10.3934/dcds.2006.14.837 |
| [19] |
Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163 |
| [20] |
Jiahong Wu. Regularity results for weak solutions of the 3D MHD equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 543-556. doi: 10.3934/dcds.2004.10.543 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]