Counterexamples to Parker's theorem (Q2737917)

The author derives two families of exact global solutions which correspond to plasma equilibrium (in stationary MHD of ideal incompressible fluid). Although the problem has been considered some time ago due to its extreme importance, its real mathematical character has been misunderstood (according to author's opinion), and so the situation has misled even eminent specialists. Therefore, this report considers newly the problem statement for eventual applications in the field of astrophysical jets and solar prominences, but, as a matter of fact, the clearing of the situation remains an important point for the theory as well. So, the global equilibria are sought, it is assumed that the magnetic field, the current and the pressure are smooth and bounded in \(\mathbb{R}^3\), the first two tend to zero, the third one to a constant value, when the distance goes to infinity; all magnetic field lines and current lines are bounded in the radial variable, if one uses cylindrical coordinates. Then, it is concluded that two families of exact global axially symmetric plasma equilibrium exist, one in the whole Euclidean space, the other in the half-space \(z\geq 0\). The second family offers applications in the desired questions. The author discusses why the solutions proposed by \textit{E. N. Parker} [Astrophys. J. 174, 499-510 (1972); Cosmical magnetic fields. Clarendon Press, Oxford (1979); Spontaneous current sheets in magnetic fields, Oxford University Press, Oxford (1994)] cannot be considered valid, because of the situation that the equilibria need to be translationary symmetric. But, unfortunately, this is not the case.