The axially symmetric Navier Stokes equation in two special domains

We introduce two special 3 dimensional domains on which something concrete can be said for the regularity problem of the axially symmetric Navier Stokes equations with suitable boundary values. One is a cusp domain and the other is the exterior of a cone. For the former, the regularity problem is solved, and for the latter, slightly less than 2/3 of the problem is  solved in some sense.  We will also discuss some open questions which seem to be the next step of research. The latter case is a joint work with Li, Zijin; Pan, Xinghong; Zeng, Chulan; Yang, Xin and Zhao, Na.