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.

Résumé

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