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 (NSν) with initial data in the scaling invariant Besov space, ℬp,∞-1+3p, here we consider a similar problem for the 3-D anisotropic Navier-Stokes equations (ANSν), where the vertical viscosity is zero. In order to do so, we first introduce the Besov-Sobolev type spaces, ℬ4-12,12 and ℬ4-12,12(T). Then with initial data in the scaling invariant space ℬ4-12,12, we prove the global wellposedness for (ANSν) provided the norm of initial data is small enough compared to the horizontal viscosity. In particular, this result implies the global wellposedness of (ANSν) 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. https://www.numdam.org/item/SEDP_2005-2006____A8_0/

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