×

Asymptotics and stability for global solutions to the Navier-Stokes equations. (English) Zbl 1038.35054

Summary: We consider an a priori global strong solution to the Navier-Stokes equations. We prove it behaves like a small solution for large time. Combining this asymptotics with uniqueness and averaging in time properties, we obtain the stability of such a global solution.

MSC:

35Q30 Navier-Stokes equations
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
PDF BibTeX XML Cite
Full Text: DOI Numdam Numdam EuDML

References:

[1] On the stability of global solutions to Navier-Stokes equations in the space · Zbl 1107.35096
[2] Calcul symbolique et propagation des singularités pour LES équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4), 14, 2, 209-246, (1981) · Zbl 0495.35024
[3] Existence of weak solutions for the Navier-Stokes equations with initial data in \(L\sp p,\) Trans. Amer. Math. Soc, 318, 1, 179-200, (1990) · Zbl 0707.35118
[4] On the regularity of the bilinear term for solutions to the incompressible Navier-Stokes equations, Rev. Mat. Iberoamericana, 16, 1, 1-16, (2000) · Zbl 0965.35121
[5] Remarques sur l’existence globale pour le système de Navier-Stokes incompressible, SIAM Journal Math. Anal, 23, 20-28, (1992) · Zbl 0762.35063
[6] Théorèmes d’unicité pour le système de Navier-Stokes tridimensionnel, J. Anal. Math, 77, 27-50, (1999) · Zbl 0938.35125
[7] Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121, 2, 314-328, (1995) · Zbl 0878.35089
[8] Unicité dans \(L\sp 3(\R\sp 3)\) et d’autres espaces fonctionnels limites pour Navier-Stokes, Rev. Mat. Iberoamericana, 16, 3, 605-667, (2000) · Zbl 0970.35101
[9] Non-explosion en temps grand et stabilité de solutions globales des équations de Navier-Stokes, C. R. Acad. Sci. Paris, Sér. I Math, 334, 289-292, (2002) · Zbl 0997.35051
[10] On infinite energy solutions to the Navier-Stokes equations: global 2D existence and 3D weak-strong uniqueness, (2001) · Zbl 1027.35090
[11] On the nonstationary Navier-Stokes system, Rend. Sem. Mat. Univ. Padova, 32, 243-260, (1962) · Zbl 0114.05002
[12] Stability estimate of strong solutions for the Navier-Stokes system and its applications, Electron. J. Differential Equations (electronic), 15, 1-23, (1998) · Zbl 0912.35120
[13] Well-posedness for the Navier-Stokes equations, Adv. Math, 157, 1, 22-35, (2001) · Zbl 0972.35084
[14] Recent progress in the Navier-Stokes problem, (2002)
[15] Sur le mouvement d’un liquide visqueux remplissant l’espace, Acta Mathematica, 63, 193-248, (1934) · JFM 60.0726.05
[16] Asymptotic behavior of global solutions to the Navier-Stokes equations in \(\R\sp 3,\) Rev. Mat. Iberoamericana, 14, 1, 71-93, (1998) · Zbl 0910.35096
[17] Sur un inégalité de type Poincaré, C. R. Acad. Sci. Paris, Sér. I Math, 330, 1, 21-23, (2000) · Zbl 0953.46020
[18] Du local au global: interpolation entre données peu régulières et lois de conservation, Séminaire: Équations aux Dérivées Partielles, Exp. No. IX, 18, 2001-2002, (2002), École Polytech., Palaiseau
[19] Global stability of large solutions to the 3D Navier-Stokes equations, Comm. Math. Phys, 159, 2, 329-341, (1994) · Zbl 0795.35082
[20] personnal communication.
[21] Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type, Annales Scientifiques de l’École Normale Supérieure, 32, 769-812, (1999) · Zbl 0938.35128
[22] The equations of Navier-Stokes and abstract parabolic equations, (1985), Friedr. Vieweg & Sohn, Braunschweig
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.