Asymptotics and stability for global solutions to the Navier-Stokes equations
Annales de l'Institut Fourier, Volume 53 (2003) no. 5, p. 1387-1424
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.
On considère une solution forte et globale des équations de Navier-Stokes. On montre qu'elle se comporte comme une solution petite en temps grand. En combinant ce résultat asymptotique avec des propriétés de moyenne en temps, on obtient la stabilité d'une telle solution globale.
DOI : https://doi.org/10.5802/aif.1983
Classification:  35B35,  35B40,  76D05
Keywords: Navier-Stokes equations, large time asymptotics, stability
@article{AIF_2003__53_5_1387_0,
     author = {Gallagher, Isabelle and Iftimie, Dragos and Planchon, Fabrice},
     title = {Asymptotics and stability for global solutions to the Navier-Stokes equations},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {53},
     number = {5},
     year = {2003},
     pages = {1387-1424},
     doi = {10.5802/aif.1983},
     zbl = {1038.35054},
     mrnumber = {2032938},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2003__53_5_1387_0}
}
Gallagher, Isabelle; Iftimie, Dragos; Planchon, Fabrice. Asymptotics and stability for global solutions to the Navier-Stokes equations. Annales de l'Institut Fourier, Volume 53 (2003) no. 5, pp. 1387-1424. doi : 10.5802/aif.1983. http://www.numdam.org/item/AIF_2003__53_5_1387_0/

[1] P. Auscher; S. Dubois; P. Tchamitchian On the stability of global solutions to Navier-Stokes equations in the space (to appear in J. Math. Pures Appl.) | MR 2062638 | Zbl 02118448

[2] J.-M. Bony Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4), Tome 14 (1981) no. 2, pp. 209-246 | Numdam | MR 631751 | Zbl 0495.35024

[3] C. P. Calderón Existence of weak solutions for the Navier-Stokes equations with initial data in L p , Trans. Amer. Math. Soc, Tome 318 (1990) no. 1, pp. 179-200 | Article | MR 968416 | Zbl 0707.35118

[4] M. Cannone; F. Planchon On the regularity of the bilinear term for solutions to the incompressible Navier-Stokes equations, Rev. Mat. Iberoamericana, Tome 16 (2000) no. 1, pp. 1-16 | Article | MR 1768531 | Zbl 0965.35121

[5] J.-Y. Chemin Remarques sur l'existence globale pour le système de Navier-Stokes incompressible, SIAM Journal Math. Anal, Tome 23 (1992), pp. 20-28 | Article | MR 1145160 | Zbl 0762.35063

[6] J.-Y. Chemin Théorèmes d'unicité pour le système de Navier-Stokes tridimensionnel, J. Anal. Math, Tome 77 (1999), pp. 27-50 | Article | MR 1753481 | Zbl 0938.35125

[7] J.-Y. Chemin; N. Lerner Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differential Equations, Tome 121 (1995) no. 2, pp. 314-328 | Article | MR 1354312 | Zbl 0878.35089

[8] G. Furioli; P. G. Lemarié; - Rieusset; E. Terraneo Unicité dans L 3 ( 3 ) et d'autres espaces fonctionnels limites pour Navier-Stokes, Rev. Mat. Iberoamericana, Tome 16 (2000) no. 3, pp. 605-667 | Article | MR 1813331 | Zbl 0970.35101

[9] I. Gallagher; D. Iftimie; F. Planchon Non-explosion en temps grand et stabilité de solutions globales des équations de Navier-Stokes, C. R. Acad. Sci. Paris, Sér. I Math, Tome 334 (2002), pp. 289-292 | MR 1891005 | Zbl 0997.35051

[10] I. Gallagher; F. Planchon On infinite energy solutions to the Navier-Stokes equations: global 2D existence and 3D weak-strong uniqueness (2001) (to appear in Arch. Rat. Mech. An) | Zbl 1027.35090

[11] T. Kato; H. Fujita On the nonstationary Navier-Stokes system, Rend. Sem. Mat. Univ. Padova, Tome 32 (1962), pp. 243-260 | Numdam | MR 142928 | Zbl 0114.05002

[12] T. Kawanago Stability estimate of strong solutions for the Navier-Stokes system and its applications, Electron. J. Differential Equations (electronic), Tome 15 (1998), pp. 1-23 | MR 1629224 | Zbl 0912.35120

[13] H. Koch; D. Tataru Well-posedness for the Navier-Stokes equations, Adv. Math, Tome 157 (2001) no. 1, pp. 22-35 | Article | MR 1808843 | Zbl 0972.35084

[14] P.-G. Lemarié Recent progress in the Navier-Stokes problem (2002) (à paraître, CRC Press)

[15] J. Leray Sur le mouvement d'un liquide visqueux remplissant l'espace, Acta Mathematica, Tome 63 (1934), pp. 193-248 | Article | JFM 60.0726.05

[16] F. Planchon Asymptotic behavior of global solutions to the Navier-Stokes equations in 3 , Rev. Mat. Iberoamericana, Tome 14 (1998) no. 1, pp. 71-93 | Article | MR 1639283 | Zbl 0910.35096

[17] F. Planchon Sur un inégalité de type Poincaré, C. R. Acad. Sci. Paris, Sér. I Math, Tome 330 (2000) no. 1, pp. 21-23 | Article | MR 1741162 | Zbl 0953.46020

[18] F. Planchon Du local au global: interpolation entre données peu régulières et lois de conservation, Séminaire: Équations aux Dérivées Partielles, École Polytech., Palaiseau, Tome Exp. No. IX, 18 (2002), p. 2001-2002

[19] G. Ponce; R. Racke; T. C. Sideris; E. S. Titi Global stability of large solutions to the 3D Navier-Stokes equations, Comm. Math. Phys, Tome 159 (1994) no. 2, pp. 329-341 | Article | MR 1256992 | Zbl 0795.35082

[20] P. Tchamitchian personnal communication.

[21] M. Vishik Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type, Annales Scientifiques de l'École Normale Supérieure, Tome 32 (1999), pp. 769-812 | Numdam | MR 1717576 | Zbl 0938.35128

[22] W. Von Wahl The equations of Navier-Stokes and abstract parabolic equations, Friedr. Vieweg \& Sohn, Braunschweig (1985) | MR 832442