Global regularity for the 3D axisymmetric incompressible Hall-MHD system with partial dissipation and diffusion

In this paper, we study the Cauchy problem for the 3D incompressible axisymmetric Hall-MHD system with horizontal velocity dissipation and vertical magnetic diffusion. We obtain a unique global smooth solution of which in the cylindrical coordinate system the swirl velocity fields, the radial and the vertical components of the magnetic fields are trivial. This type of solution has been studied for the MHD system in [17], [16] and [15] and for the Hall-MHD system with total dissipation and diffusion in [11]. Some new and fine estimates are obtained in this paper to overcome the di?iculties raised from the Hall term and the loss of vertical velocity dissipation and horizontal magnetic diffusion. Finally we can show that the estimates $\\int_{O}^{T}$ ??u(t)?L? dt and $\\int_{O}^{T}$ ??b(t)?L? dt are finite in a priori way and hence obtainthe global well-posedness to the system under considered.