Meissner states of type II superconductors. (English) Zbl 1410.82039
Author’s abstract: This paper is concerned with the mathematical theory of Meissner states of a bulk superconductor of type II, which occupies a bounded domain \(\Omega\) in \(\mathbb{R}^3\) and is subjected to an applied magnetic field below the critical field \(H_{\mathrm{S}}\). A Meissner state is described by a solution \((f,\mathbf{A})\) of a nonlinear partial differential system called Meissner system, where \(f\) is a positive function on \(\Omega\) which is equal to the modulus of the order parameter, and \(\mathbf{A}\) is the magnetic potential defined on the entire space such that the inner trace of the normal component on the domain boundary \(\partial\Omega\) 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 values of the Ginzburg-Landau parameter \(\kappa\). It is shown that the Meissner solution is smooth in \(\Omega\), however, the regularity of the magnetic potential outside \(\Omega\) can be rather poor. This observation leads to the ides of decomposition of the Meissner system into two problems, a boundary value problem in \(\Omega\) and an exterior problem outside of \(\Omega\). We show that the solutions of the boundary value problem with fixed boundary data converges uniformly on \(\Omega\) as \(\kappa\) tends to \(\infty \), where the limit field of the magnetic potential is a solution of a nonlinear curl system. This indicates that the magnetic potential part \(\mathbf{A}\) of the solution \((f,\mathbf{A})\) of the Meissner system, which has same tangential component of curl \(\mathbf{A}\) on \(\partial\Omega\), converges to a solution of the curl system as \(\kappa\) increases to infinity, which verifies that the curl system is indeed the correct limit of the Meissner system in the case of three dimensions.
Reviewer: Anthony D. Osborne (Keele)
MSC:
| 82D55 | Statistical mechanics of superconductors |
| 35B25 | Singular perturbations in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B45 | A priori estimates in context of PDEs |
| 35J47 | Second-order elliptic systems |
| 35J57 | Boundary value problems for second-order elliptic systems |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q56 | Ginzburg-Landau equations |
Keywords:
Ginzburg-Landau system; superconductivity; Meissner state; superheating field; elliptic equations; quasilinear system; asymptotic behaviorReferences:
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