Abstract: The 3D incompressible Navier-Stokes with dissipation in a single direction in R^3 is not well-understood, and the small data global well-posedness problem remains open. However, when this Navier-Stokes is coupled with the equation of the magnetic field, the resulting MHD system has been shown to be globally well-posed near a background magnetic field. In addition, the perturbation decays with 2D heat-like behaviors. This reveals the stabilizing and damping effect of the magnetic field. Even in domains with simple boundary such as the half space, these mechanisms persist and allow global-in-time uniform bounds, well-posedness and decay near a background magnetic field.
