Global solution of 3D axially symmetric nonhomogeneous incompressible MHD equations

We construct a global smooth solution of the nonhomogeneous (density not to be a constant) incompressible axially symmetric MHD equations in three space dimensions. Here we do not require some components of velocity field and magnetic field to be zero, and no other regularity assumptions. Firstly, we propose a new one-dimensional model which approximates the incompressible MHD equations. The nonlinear structure of the one-dimensional has some very interesting properties. Secondly, we find that this one-dimensional model is the one which can construct a family solutions of the three-dimensional axially symmetric MHD equations, what's more we prove the global regularity of the one-dimensional model for a family of initial data. Finally, we prove global regularity of the three-dimensional MHD equations by the special nonlinear structure of one-dimensional model.