The role of oscillations in the global wellposedness of the 3-D incompressible anisotropic Navier-Stokes equations
Séminaire Équations aux dérivées partielles (Polytechnique) (2005-2006), Talk no. 8, 18 p.

Corresponding to the wellposedness result [2] for the classical 3-D Navier-Stokes equations (NS ν ) with initial data in the scaling invariant Besov space, p, -1+3 p , here we consider a similar problem for the 3-D anisotropic Navier-Stokes equations (ANS ν ), where the vertical viscosity is zero. In order to do so, we first introduce the Besov-Sobolev type spaces, 4 -1 2,1 2 and 4 -1 2,1 2 (T). Then with initial data in the scaling invariant space 4 -1 2,1 2 , we prove the global wellposedness for (ANS ν ) provided the norm of initial data is small enough compared to the horizontal viscosity. In particular, this result implies the global wellposedness of (ANS ν ) with high oscillatory initial data.

@article{SEDP_2005-2006____A8_0,
     author = {Chemin, Jean-Yves and Zhang, Ping},
     title = {The role of oscillations in the global wellposedness of the 3-D incompressible anisotropic Navier-Stokes equations},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique)},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2005-2006},
     note = {talk:8},
     mrnumber = {2276074},
     language = {en},
     url = {http://www.numdam.org/item/SEDP_2005-2006____A8_0}
}
Chemin, Jean-Yves; Zhang, Ping. The role of oscillations in the global wellposedness of the 3-D incompressible anisotropic Navier-Stokes equations. Séminaire Équations aux dérivées partielles (Polytechnique) (2005-2006), Talk no. 8, 18 p. http://www.numdam.org/item/SEDP_2005-2006____A8_0/

[1] J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Annales de l’École Normale Supérieure, 14, 1981, pages 209-246. | Numdam | Zbl 0495.35024

[2] M. Cannone, Y. Meyer and F. Planchon, Solutions autosimilaires des équations de Navier-Stokes, Séminaire "Équations aux Dérivées Partielles de l’École Polytechnique", Exposé VIII, 1993–1994. | Numdam | Zbl 0882.35090

[3] J.-Y. Chemin, Théorèmes d’unicité pour le système de Navier-Stokes tridimensionnel, Journal d’Analyse Mathématique, 77, 1999, pages 27–50. | Zbl 0938.35125

[4] J.-Y. Chemin, Localization in Fourier space and Navier-Stokes system, Phase Space Analysis of Partial Differential Equations, Proceedings 2004, CRM series, Pisa, pages 53-136. | MR 2144406 | Zbl 1081.35074

[5] J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Basics of Mathematical Geophysics, Preprint of CMLS, École polytechnique, 2004. | MR 2228849

[6] J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Fluids with anisotropic viscosity, Modélisation Mathématique et Analyse Numérique, 34, 2000, pages 315-335. | Numdam | MR 1765662 | Zbl 0954.76012

[7] J.-Y. Chemin and P. Zhang, On the global wellposedness to the 3-D incompressible anisotropic Navier-Stokes equations, to appear in | MR 2300252 | Zbl 1132.35068

[8] T.-M. Fleet, Differential Analysis, Cambridge University Press, 1980. | MR 561908 | Zbl 0442.34002

[9] H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Archiv for Rational Mechanic Analysis, 16, 1964, pages 269–315. | MR 166499 | Zbl 0126.42301

[10] D. Iftimie, A uniqueness result for the Navier-Stokes equations with vanishing vertical viscosity, SIAM Journal of Mathematical Analysis, 33, 2002, pages 1483–1493. | MR 1920641 | Zbl 1011.35105

[11] H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations. Advances in Mathematics, 157, 2001, pages 22–35. | MR 1808843 | Zbl 0972.35084

[12] M. Paicu, Equation anisotrope de Navier-Stokes dans des espaces critiques, Revista Matematica Iberoamericana, 21, 2005, pages 179–235. | MR 2155019 | Zbl 1110.35060

[13] M. Vishik, Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type, Annales de l’École Normale Supérieure, 32, 1999, pages 769-812. | Numdam | Zbl 0938.35128