Bounds and New Approaches for the 3D MHD Equations

Magnetohydrodynamics (MHD), the science of the motion of an electrically conducting fluid in the presence of a magnetic field, consists essentially of the interaction between the fluid velocity and the magnetic field. Electric currents induced in the fluid as a result of its motion modify the field; at the same time their flow in the magnetic field leads to mechanical forces which modify the motion. The equations of MHD are the usual hydrodynamic and electromagnetic equations, modified to take this interaction into account. If u is the fluid velocity, b the magnetic field, and j the current density, then