In this paper, we consider the global wellposedness of the 3-D incompressible anisotropic Navier-Stokes equations with initial data in the critical Besov-Sobolev type spaces. This is partially joint work with J. Y. Chemin, Guilong GUI and M. Paicu.

I shall talk about some regularity results for the ultraparabolic equation, in particular, the C^{alpha}$ regularity of weak solutions. The problem arises from the Prandtl's boundary layer system under the Crocco transformation. I shall also report some recent results on the backward uniqueness of ultraparabolic equations.