Axially symmetric incompressible MHD in three dimensions

In this paper, we study the property of solutions to axially symmetric incompressible MHD equations in three dimensions. Firstly, we present the three-dimensional axially symmetric incompressible MHD equations. We propose a new one-dimensional model that approximates the MHD equations along the symmetric axis. To our surprise, this one-dimensional model can construct a family of exact solutions to the three-dimensional MHD equations. Secondly, we give a family of particular solutions to the three-dimensional and the prior estimates of one-dimensional MHD equations. Finally, we construct a family of global smooth solutions to the three-dimensional MHD equations by applying the one-dimensional solutions.