Global existence of 3D MHD equations with mixed partial dissipation and magnetic diffusion

This article considers the existence of global smooth solutions to the Cauchy problem in 3D incompressible magnetohydrodynamic(MHD) flows with mixed partial dissipation and magnetic diffusion. We prove that if the initial data satisfy ‖u0‖H1+‖b0‖H1≤ϵ‖u0‖H1+‖b0‖H1≤ϵ, where ϵϵ is a sufficiently small positive number, then the 3D MHD equations with mixed partial dissipation and magnetic diffusion admit global smooth solutions.