Global strong solutions to the anisotropic three-dimensional incompressible magnetohydrodynamic system

In this article, we consider the Cauchy problem of the three-dimensional incompressible magnetohydrodynamic (MHD) system with only horizontal dissipation and vertical magnetic diffusion near the equilibrium. We establish the global a priori estimate of the smooth solution for this system, with initial data (u0,B0-e1)$$ \left({u}_0,{B}_0-{e}_1\right) $$ sufficiently small in H2$$ {H} circumflex 2 $$, by using the special structure of the nonlinear terms and the anisotropic Sobolev inequality.