Meissner states of type II superconductors

Abstract

This paper concerns mathematical theory of Meissner states of a bulk superconductor of type II, which occupies a bounded domain Ω in R 3 and is subjected to an applied magnetic field below the critical field HS. A Meissner state is described by a solution (f, A) of a nonlinear partial differential system called Meissner system, where f is a positive function on Ω which is equal to the modulus of the order parameter, and A is the magnetic potential defined on the entire space such that the inner trace of the normal component on the domain boundary ∂Ω vanishes. Such a solution is called a Meissner solution. Various properties of the Meissner solutions are examined, including regularity, classification and asymptotic behavior for large value of the Ginzburg-Landau parameter κ. It is shown that the Meissner solution is smooth in Ω, however the regularity of the magnetic potential outside Ω can be rather poor. This observation leads to the ides of decomposition of the Meissner system into two problems, a boundary value problem in Ω and an exterior problem outside of Ω. We show that the solutions of the boundary value problem with fixed boundary data converges uniformly on Ω as κ tends to ∞, where the limit field of the magnetic potential is a solution of a nonlinear curl system. This indicates that, the magnetic potential part A of the solution (f, A) of the Meissner system, which has same tangential component of curl A on ∂Ω, converges to a solution of the curl system as κ increases to infinity, which verifies that the curl system is indeed the correct limit of the Meissner system in the case of three dimensions.Published in:

Variational and operator methods for Maxwell-Stokes system

“…The restriction for the domain to be simply-connected and without holes is removed in [35], hence the results in [10] remain true for a general bounded domain. More precise results of the location of concentration points is proved in [70], and the full model of Meissner states are studies in [44,50]. These results for (130) give the corresponding results for (129).…”

“…We show that ζ is equal to a constant number. From (50) and 55, for any w ∈ W 1,p t0 (Ω, div 0) we have…”

“…In the same manner we define T φ,c for φ ∈ L p (Ω) and c ∈ R (see (50)). If furthermore φ has zero trace on ∂Ω then we may write T φ as a "gradient", T φ = ∇φ, which is understood as a functional on W 1,p t0 (Ω, R 3 ) .…”

“…For the Maxwell type equations, Schauder regularity of weak solutions has been established for many quasilinear models, see [10,46,47,35,50] for the model of Meissner states of superconductivity, [48,49] for the nonlinear Maxwell equations.…”