Global magnetofluidostatic fields (an unsolved PDE problem) (Q1090570)

This paper is concerned with the problem of finding possible magnetostatic solutions in a fluid where the pressure force (-\(\nabla P)\) is exactly balanced by the ponderomotive force \((J\times B=(\nabla \times H)\times B)\). Several difficulties are cataloged and it is concluded that the general solution is still an open mathematical question, and still unexplored as a singular-perturbation problem. It is pointed out that this problem is of strong relevance in magnetohydrodynamics (i.e., where inertia and viscous terms are neglected). The paper fails to make mention of, but it is interesting to speculate on, the meaning of any such solutions on general theory since most known static solutions are unstable in the MFD sense. Further it might be suspected that the problem as stated is ambiguous since the nature of the force balance across the system global boundary and the source of the current (\(\nabla \times H)\) are both important in determining the nature of the static solution and ultimate stability.