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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 sufficiently small in , by using the special structure of the nonlinear terms and the anisotropic Sobolev inequality.
Han et al. (Tue,) studied this question.