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Séminaire Équations aux dérivées partielles (Polytechnique)

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Jean-Yves Chemin; Ping Zhang
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), Exp. No. 8, 18 p.
Article PDF | Reviews MR 2276074

Résumé - Abstract

Corresponding to the wellposedness result [2] for the classical 3-D Navier-Stokes equations $(NS_\nu )$ with initial data in the scaling invariant Besov space, ${\cal B}^{-1+\frac{3}{p}}_{p,\infty },$ here we consider a similar problem for the 3-D anisotropic Navier-Stokes equations $(ANS_\nu ),$ where the vertical viscosity is zero. In order to do so, we first introduce the Besov-Sobolev type spaces, ${\cal B}^{-\frac{1}{2},\frac{1}{2}}_4$ and ${\cal B}^{-\frac{1}{2},\frac{1}{2}}_4(T).$ Then with initial data in the scaling invariant space ${\cal B}^{-\frac{1}{2},\frac{1}{2}}_4,$ we prove the global wellposedness for $(ANS_\nu )$ provided the norm of initial data is small enough compared to the horizontal viscosity. In particular, this result implies the global wellposedness of $(ANS_\nu )$ with high oscillatory initial data.

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