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This paper focuses on the 3D incompressible magnetohydrodynamic (MHD)equations with mixed partial dissipation and magnetic diffusion. Our mainresult assesses the global stability of perturbations near the steady solutiongiven by a background magnetic field. The stability problem on the MHDequations with partial or no dissipation has attracted considerable interestsrecently and there are substantial developments. The new stability resultpresented here is among the very few stability conclusions currently availablefor ideal or partially dissipated MHD equations. As a special consequence ofthe techniques introduced in this paper, we obtain the small data globalwell-posedness for the 3D incompressible Navier-Stokes equations withoutvertical dissipation.

Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equilibrium