3. Grad-Rubin approach
Let us follow the approach that was proposed by Grad and Rubin (1958). The previous underlying mixed elliptic-hyperbolic structure of the system of equations is exploited by introducing the following sequences of hyperbolic and elliptic linear BVPs:
that is given by (Aly 1989):
These sequences of problems must be proved to converge towards the solution of the original BVP defined by Eqs. (1) and (2) provided with the set of boundary conditions given by Eqs. (3),5). One can use them to address theoretical issues such as i) existence of solution ii) uniqueness iii) continuity of the solution with the respect to the boundary conditions. These three points define a well-posed BVP in the sense of Hadamard (1932) and has been discussed for other BVP associated to extrapolation methods (Low & Lou 1991, Amari et al. 1998). Note that the last point is important because of the presence of errors in the measurements of the photospheric magnetic field and of the possible non-force-free character of the field at the photospheric level, where pressure and dynamic forces can play a non-negligible role (Aly 1989, McClymont et al. 1997). Of course those three points depend on the functional space in which one seeks the solution, and in particular on the the regularity of the solution (Amari 1991).
Bineau (1972), considered this BVP in the Holder functional spaces (set of functions sufficiently regular and whose derivatives are also regular enough, Brezis 1983). The BVP is then proved to be well-posed when . However, this proof rests on the following assumptions: (i) The domain is bounded. (ii) The field as well as have a simple magnetic topology (then they must not vanish in ). It is however possible to show the existence of a solution for bounded, in more general spaces (when (), that is in a functional space such that solution may admit separatrices surfaces, null points, and current sheets, (Boulmezaoud, Amari & Maday, in preparation). Uniqueness of the solution has not yet been proved.
© European Southern Observatory (ESO) 1999
Online publication: October 14, 1999