Bean's critical-state model as the p → limit of an evolutionary p -Laplacian equation

We consider magnetization of type II superconductors characterized by a multi valued current voltage relation the Bean model and show that for the longitudinal and thin lm con gurations the problems are equivalent to similar evolutionary variational inequalities with a gradient constraint It is proved that the unique solutions to these inequalities are the p limits of the solutions to the evo lutionary p Laplacian equations that appear in the corresponding magnetization models obeying the power current voltage law with exponent p Introduction The Bean critical state model provides a description for the magnetization of type II su perconductors in a nonstationary external magnetic eld The model was rst formulated for the simplest con guration of a cylindrical superconductor in a parallel eld see Since then more complicated cases have also been considered in particular a very thin superconducting lm in a perpendicular external eld see and the references therein Phenomenologically the problem can be understood as a nonlinear eddy current problem In accordance with the Faraday law of electromagnetic induction the eddy currents in a conductor are driven by the electric elds induced by time variations of the magnetic ux In an ordinary conductor the vectors of the electric eld and the current density are usually related by the linear Ohm law Type II superconductors are instead characterized in the Bean model by a highly nonlinear current voltage relation This non linearity gives rise to an interesting free boundary problem which is considered here for the two speci c geometrical con gurations mentioned above a long cylinder in a parallel magnetic eld and a thin lm in a perpendicular eld Acknowledges travel support from Marks and Spencer Ltd Present address CEEP Blaustein Inst for Desert Research Ben Gurion University of the Negev Sede Boqer Campus Israel In these cases the electric eld e inside the isotropic superconductor has the same direction as the current density j and the superconducting material may be characterized by a scalar current voltage law This nonlinear constitutive relation is given in the Bean model by a multivalued monotone graph jej if jjj if jjj if jjj Here we have adopted units in which the critical current density jc The magne tization model with this current voltage law is equivalent to an evolutionary variational inequality see and such a formulation is convenient for both the numerical approxi mation and theoretical study of these magnetization problems In simple cases the solution to the Bean model can be found analytically see e g the two examples in the next section Physicists however usually approximate by a smooth function in order to sim plify the numerical discretization or to account for the thermally activated creep of the magnetic ux see The power law approximation jej jjj for a xed large p R is the most often adopted This approximation leads to evolu tionary equations involving the p Laplacian operator and it was assumed in the physical literature that their solutions converge to the Bean model solution as p In this paper we study the behaviour of the solutions to these evolutionary equations for two geometrical con gurations and prove rigorously that the convergence does indeed take place in each case to the unique solution of the corresponding evolutionary variational inequality equivalent to the Bean critical state model for that con guration For long cylinders in a parallel eld the variational inequality problem can be written in terms of the magnetic eld and involves a gradient constraint This problem is similar to that arising in another critical state model the sandpile growth model see Recently Aronsson Evans and Wu have shown that the sandpile growth model can be obtained as the p limit of the Cauchy problem for an evolutionary p Laplacian equation We partially adopt their techniques in our consideration of the corresponding limits of the similar boundary value problems in superconductivity It should be noted that the similarity between the magnetization of type II superconductors and the growth of sandpiles is well known see In the case of a thin superconducting lm placed into a perpendicular magnetic eld a variational inequality with the same gradient constraint as in the cylindrical case can be derived for the stream function of a divergence free two dimensional d sheet current density This evolutionary variational inequality is implicit with respect to the time derivative and we prove that it is the p limit of an implicit evolutionary equation involving the p Laplacian operator In the next section we derive variational formulations for the power law and Bean magnetization problems for these two speci c geometrical con gurations Although the two con gurations lead to di erent mathematical problems these can be regarded as two special cases of a more general evolutionary problem involving the p Laplacian Therefore in section we analyse the well posedness of this more general problem and study its limit as p Variational formulation of the models Let R be a bounded connected domain with a Lipschitz boundary If is not simply connected we allow it to have a nite number of holes i i I with i being a bounded domain with a connected boundary i We set