The role of oscillations in the global wellposedness of the 3-D incompressible anisotropic Navier-Stokes equations
Chemin, Jean-Yves1 ; Zhang, Ping2
1 Laboratoire J.-L. Lions, Case 187 Université Pierre et Marie Curie, 75230 Paris Cedex 05, FRANCE
2 Academy of Mathematics & Systems Science, CAS Beijing 100080, CHINA.
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2005-2006), Exposé 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, ℬ 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, ℬ 4 -1 2,1 2 and ℬ 4 -1 2,1 2 (T). Then with initial data in the scaling invariant space ℬ 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.

Chemin, Jean-Yves 1 ; Zhang, Ping 2

1 Laboratoire J.-L. Lions, Case 187 Université Pierre et Marie Curie, 75230 Paris Cedex 05, FRANCE
2 Academy of Mathematics & Systems Science, CAS Beijing 100080, CHINA. 
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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) dit aussi "Séminaire Goulaouic-Schwartz" (2005-2006), Exposé no. 8, 18 p. http://archive.numdam.org/item/SEDP_2005-2006____A8_0/

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