| Abstract: |
| The small data global well-posedness of the 3D incompressible Navier-Stokes equations with only one-directional dissipation in the whole space remains an outstanding open problem. Motivated by this Navier-Stokes open problem and by experimental observations on the stabilizing effects of background magnetic fields, we investigate the global well-posedness, the stability and large-time behavior of a special 3D magnetohydrodynamic (MHD) system with only one-directional velocity dissipation and horizontal magnetic diffusion near a background magnetic field in the whole space. Firstly, by discovering the mathematical mechanism of the experimentally observed stabilizing effect and introducing several new techniques to unearth the hidden structure in the nonlinearity, we overcome the derivative loss difficulties and solve the desired global well-posedness and stability problem. Furthermore, by initiating new strategies and developing innovative tools for stability and large-time behavior problems on anisotropic models, we improve the stability to the weaker Sobolev setting . Meanwhile, explicit decay rates are also obtained. |
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