Asymptotics and stability for global solutions to the Navier-Stokes equations

Isabelle Gallagher[1]; Dragos Iftimie[2]; Fabrice Planchon[3]

  • [1] École Polytechnique, Centre de Mathématiques, UMR 7640, 91128 Palaiseau (France)
  • [2] Université de Rennes 1, IRMAR, UMR 6625, Campus de Beaulieu, 35042 Rennes (France)
  • [3] Université Paris 13, Institut Galilée, Laboratoire d’Analyse, Géométrie & Applications, UMR 7539, avenue J.-B. Clément, 93430 Villetaneuse (France)

Annales de l’institut Fourier (2003)

  • Volume: 53, Issue: 5, page 1387-1424
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Gallagher, Isabelle, Iftimie, Dragos, and Planchon, Fabrice. "Asymptotics and stability for global solutions to the Navier-Stokes equations." Annales de l’institut Fourier 53.5 (2003): 1387-1424. <http://eudml.org/doc/116076>.

@article{Gallagher2003,
abstract = {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.},
affiliation = {École Polytechnique, Centre de Mathématiques, UMR 7640, 91128 Palaiseau (France); Université de Rennes 1, IRMAR, UMR 6625, Campus de Beaulieu, 35042 Rennes (France); Université Paris 13, Institut Galilée, Laboratoire d’Analyse, Géométrie & Applications, UMR 7539, avenue J.-B. Clément, 93430 Villetaneuse (France)},
author = {Gallagher, Isabelle, Iftimie, Dragos, Planchon, Fabrice},
journal = {Annales de l’institut Fourier},
keywords = {Navier-Stokes equations; large time asymptotics; stability; infinite energy weak solutions; strong solutions; paradifferential calculus},
language = {eng},
number = {5},
pages = {1387-1424},
publisher = {Association des Annales de l'Institut Fourier},
title = {Asymptotics and stability for global solutions to the Navier-Stokes equations},
url = {http://eudml.org/doc/116076},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Gallagher, Isabelle
AU - Iftimie, Dragos
AU - Planchon, Fabrice
TI - Asymptotics and stability for global solutions to the Navier-Stokes equations
JO - Annales de l’institut Fourier
PY - 2003
PB - Association des Annales de l'Institut Fourier
VL - 53
IS - 5
SP - 1387
EP - 1424
AB - 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.
LA - eng
KW - Navier-Stokes equations; large time asymptotics; stability; infinite energy weak solutions; strong solutions; paradifferential calculus
UR - http://eudml.org/doc/116076
ER -

References

top
  1. P. Auscher, S. Dubois, P. Tchamitchian, On the stability of global solutions to Navier-Stokes equations in the space Zbl1107.35096MR2062638
  2. J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4) 14 (1981), 209-246 Zbl0495.35024MR631751
  3. C. P. Calderón, Existence of weak solutions for the Navier-Stokes equations with initial data in , Trans. Amer. Math. Soc 318 (1990), 179-200 Zbl0707.35118MR968416
  4. M. Cannone, F. Planchon, On the regularity of the bilinear term for solutions to the incompressible Navier-Stokes equations, Rev. Mat. Iberoamericana 16 (2000), 1-16 Zbl0965.35121MR1768531
  5. J.-Y. Chemin, Remarques sur l'existence globale pour le système de Navier-Stokes incompressible, SIAM Journal Math. Anal 23 (1992), 20-28 Zbl0762.35063MR1145160
  6. J.-Y. Chemin, Théorèmes d'unicité pour le système de Navier-Stokes tridimensionnel, J. Anal. Math 77 (1999), 27-50 Zbl0938.35125MR1753481
  7. J.-Y. Chemin, N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differential Equations 121 (1995), 314-328 Zbl0878.35089MR1354312
  8. G. Furioli, P. G. Lemarié, - Rieusset, E. Terraneo, Unicité dans et d’autres espaces fonctionnels limites pour Navier-Stokes, Rev. Mat. Iberoamericana 16 (2000), 605-667 Zbl0970.35101MR1813331
  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 334 (2002), 289-292 Zbl0997.35051MR1891005
  10. I. Gallagher, F. Planchon, On infinite energy solutions to the Navier-Stokes equations: global 2D existence and 3D weak-strong uniqueness, (2001) Zbl1027.35090
  11. T. Kato, H. Fujita, On the nonstationary Navier-Stokes system, Rend. Sem. Mat. Univ. Padova 32 (1962), 243-260 Zbl0114.05002MR142928
  12. T. Kawanago, Stability estimate of strong solutions for the Navier-Stokes system and its applications, Electron. J. Differential Equations (electronic) 15 (1998), 1-23 Zbl0912.35120MR1629224
  13. H. Koch, D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math 157 (2001), 22-35 Zbl0972.35084MR1808843
  14. P.-G. Lemarié, Recent progress in the Navier-Stokes problem, (2002) Zbl1034.35093
  15. J. Leray, Sur le mouvement d'un liquide visqueux remplissant l'espace, Acta Mathematica 63 (1934), 193-248 
  16. F. Planchon, Asymptotic behavior of global solutions to the Navier-Stokes equations in , Rev. Mat. Iberoamericana 14 (1998), 71-93 Zbl0910.35096MR1639283
  17. F. Planchon, Sur un inégalité de type Poincaré, C. R. Acad. Sci. Paris, Sér. I Math 330 (2000), 21-23 Zbl0953.46020MR1741162
  18. F. Planchon, Du local au global: interpolation entre données peu régulières et lois de conservation, Séminaire: Équations aux Dérivées Partielles Exp. No. IX, 18 (2002), 2001-2002, École Polytech., Palaiseau 
  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 159 (1994), 329-341 Zbl0795.35082MR1256992
  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 32 (1999), 769-812 Zbl0938.35128MR1717576
  22. W. von Wahl, The equations of Navier-Stokes and abstract parabolic equations, (1985), Friedr. Vieweg & Sohn, Braunschweig MR832442

NotesEmbed ?

top

You must be logged in to post comments.