In this talk I will first report a Liouville type Theorem for the axi-symmetric Navier-Stokes equations. The proof relies on the extra conservation law which is obtained by using the convection term. In the second part, I will talk about a 3D model equation for Navier-Stokes equations. This 3D model is obtained by ignoring the convection terms in a reformulated axi-symmetric Navier-Stokes equations. The 3D model possesses the properties that the energy law is satisfied, it is incompressible, the nonlinearity is nonlocal, etc., which are similar to those for the Naiver-Stokes equations. Consequently, in most cases, whatever you can prove for the Navier-Stokes equations, you may also have similar results for our 3D model. Then I will report that the 3D inviscid model can develop finite time singularities starting from smooth initial data with finite energy. The above two parts suggest that convection terms may play a positive role for the global regularity theory of the incompressible Euler and Navier-Stokes equations. At last, I will report a very recent result of constructing a family of large solutions for the 3D Navier-Stokes equations which is obtained by making use of the convection to break the scaling. Those are a series work joint with a number of collaborators: professor Thomas Y. Hou from Caltech, professor Congming Li from University of Colorado, professor Fang-hua Lin from Courant Institute, professor Shu Wang from Beijing University of Technology, professor Qi S. Zhang from University of California at Riverside, professor Yi Zhou from Fudan university and Dr. Chen Zou from Beking University.