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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. |
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