MAXWELL QUASI-VARIATIONAL INEQUALITIES IN SUPERCONDUCTIVITY

This paper is devoted to the mathematical modeling and analysis of a hyperbolic Maxwell quasi-variational inequality (QVI) for the Bean-Kim superconductivity model with temperature and magnetic field dependence in the critical current. Our analysis relies on local (resp. global) boundedness and local (resp. global) Lipschitz continuity assumptions on the critical current with respect to the temperature (resp. magnetic field). Emerging from the Euler time discretization, we analyze the corresponding H(curl)-elliptic QVI and prove its existence using a fixed-point argument in combination with techniques from variational inequalities and Maxwell’s equations. Based on the existence result for the H(curl)-elliptic QVI, we examine the stability and convergence of the Euler scheme, which serve as our fundament for the global well-posedness of the governing hyperbolic Maxwell QVI. Mathematics Subject Classification. 35L85, 35Q60. Received January 13, 2021. Accepted June 15, 2021.


Introduction
Ever since the discovery of superconductivity by Heike Kamerlingh Onnes in 1911, various modern applications and key technologies have been developed. Among many other profound applications, we mention magnetic resonance imaging, magnetic confinement fusion, and magnetic levitation. Such technological advances are made possible by superconductors due to their fundamental properties of vanishing electrical resistance and expulsion of applied magnetic fields (Meissner effect) occurring when the temperature is cooled below the critical temperature. A prominent critical-state model describing the irreversible magnetization process in high-temperature superconductivity was proposed by Bean [5,6]. His model postulates a nonlinear and non-smooth constitutive relation between the current density and the electric field through the so-called critical current as follows: (B1) the current density strength | | cannot exceed the critical current (B2) the electric field vanishes if | | is strictly less than (B3) the electric field is parallel to the current density .
Shortly after the publication by Bean [5], Kim et al. [13] revealed and reported on the magnetic field dependency in the critical current density = ( ).
The Bean-Kim model governed by the eddy current equations leads to a parabolic quasi-variational inequality (QVI). Prigozhin [20] was the first to introduce this formulation. Barrett and Prigozhin [3,4] examined the associated QVI in a scalar two-dimensional (2D) setting and its dual formulation. For the analysis of general parabolic and elliptic QVI problems with gradient and curl constraints, we refer to Rodrigues and Santos [21,22] and Miranda et al. [16]. All these contributions take into account the eddy current approximation of the full Maxwell formulation leading to problems with a parabolic character. The analysis of Bean's critical-state model with displacement current goes back to [12].
This paper is a continuation of the recent papers [32,33] on hyperbolic Maxwell variational inequalities (VI), including those arising in high-temperature superconductivity and electromagnetic shielding (cf. [14,27,28]). The goal of the present paper is to explore hyperbolic Maxwell QVI arising from the Bean-Kim model (B1)-(B3) with magnetic field and temperature dependence in the critical current = ( , ). In particular, temperature effects are included due to its central importance in the superconductivity phenomena. As reported in [2], the temperature dependence in the critical current of the Y-Ba-Cu-O bulk superconductor exhibits a continuous and piecewise smooth structure ( [2], Fig. 2). More precisely, it features a linear behaviour of the type ( − ), if the temperature is sufficiently smaller than (see Tab. 1). If is close to , then a nonlinear behaviour of the type (1 − ) 3/2 is observed, and the critical current vanishes if ≥ . This behaviour is in agreement with the theoretical model of granular superconductors [8]. Similarly, Deutscher and Müller [9] reported on a temperature dependence of the type (1 − ) 2 in the case of ≈ for the critical currents of high-temperature oxides.
The first part of this paper is devoted to the following nonlinear PDE-system: for a.e. ∈ Ω ( ) · ( ) = ( , ( ), ℎ( ))| ( )| for a.e. ∈ Ω, (1.2) where f , g, and are given data. The variational formulation of (1.2) in terms of (see Cor. 3.3) is given by the following (curl)-elliptic QVI: Find ∈ 0 (curl) such that where the bilinear form : 0 (curl) × 0 (curl) → R is defined as in (3.3). To derive an existence result for (1.2), we first drop out the magnetic field dependency in the critical current . This leads to a complementaritytype problem, which we shall study through the theory of variational inequalities. Then, on the basis of the proposed complementarity-type problem, we formulate (1.2) as a fixed-point problem and show its existence (Thm. 3.2) by means of the Maxwell compactness embedding theory [19,26] along with the Schauder fixed-point theorem.
After deriving a well-posedness result for (1.2), we consider the time-discrete problem (P ) associated with (1.1) based on the implicit Euler scheme (Rothe method). While the existence of (P ) is covered by the developed existence result for (1.2), the uniqueness is obtained if the time step is sufficiently small. We investigate the stability analysis of the resulting time-discrete magnetic and electric fields, including their difference quotients (Lems. 4.5 and 4.6). Differently from Assumption 2.2 of [27], our stability analysis does not rely on any compatibility condition for the initial data. We circumvent this issue by introducing an auxiliary current density (4.2) and initial difference quotients (4.3), that preserve the pivotal QVI structure at the initial time (4.4). This construction allows us to prove stability and convergence of (P ) leading to our final result (Cor. 5.2) on the well-posedness for (QVI) for all ∈ 1 ((0, ), 2 (Ω)), ∈ 1 ((0, ), 2 (Ω)) ∩ ([0, ], ∞ (Ω)), and ( 0 , 0 ) ∈ 0 (curl) × (Ω) without any compatibility assumption.

Preliminaries
For a given Hilbert space , we use the notation ‖ · ‖ and (·, ·) for a standard norm and a standard scalar product in . By * we denote the dual space of . If is continuously embedded in another normed linear space , then we write ˓→ for the associated injection. A bold typeface is used to indicate a threedimensional vector function or a Hilbert space of three-dimensional vector functions. The main Hilbert space for our analysis is where the curl -operator is understood in the sense of distributions. As usual, ∞ 0 (Ω) stands for the space of all infinitely differentiable three-dimensional vector functions with compact support contained in Ω. We denote the closure of ∞ 0 (Ω) with respect to the (curl)-topology by 0 (curl). It is well-known that the Hilbert space 0 (curl) admits the following characterization (cf. [32], Appendix A): where : (curl) → − 1 2 ( Ω) denotes the tangential trace (cf. [11], Thm. 2.11). We note that ( ) = 0 generalizes the boundary condition × = 0 on Ω. Another important Hilbert space used in our analysis is For an almost everywhere positive function ∈ ∞ (Ω), we use the notation 2 (Ω) for the weighted 2 (Ω)-space endowed with the weighted scalar product ( ·, ·) 2 (Ω) . Let us now formulate the required regularity assumption on the electric permittivity : Ω → (0, ∞), the magnetic permeability : Ω → (0, ∞), and the critical current : for a.e. ∈ Ω and all ( 1 , ℎ 1 ), Note that (A3) and (A4) require the global boundedness and the global Lipschitz continuity of : Ω × R × R 3 → [0, ∞) with respect to the third component (magnetic field). An example for the magnetic field dependency satisfying (A3) and (A4) is for some positive constants 1 , 2 , 3 > 0 and some exponent > 1. The mapping (2.3) is obviously globally bounded. Furthermore, the Lipschitz continuity holds true as its derivative is globally bounded. As confirmed by physical measurement [7], such a model (2.3) is reasonable for describing the magnetic field dependency in critical currents of certain superconductors.

Well-posedness for (EQVI)
We start our investigation by examining the following complementarity-type problem.
In conclusion, the triple ( , ℎ, Let us prove the a priori estimate (3.2). First, equation (3.5) implies from which it follows that Also, the inequalities (3.13) and (3.11) together with (A3) imply (3.15) From (3.14) and (3.15), we come to the conclusion that From (3.16) and (3.17) together with (2.1), we obtain that On the other hand, the properties (3.18) and (3.19) imply Altogether, we see that̃︀ ∈ 0 (curl) satisfies But, we know that ∈ 0 (curl) is the unique solution of the above variational inequality, and hencẽ︀ = .

Time-discrete problem
This section is devoted to the analysis of the time-discrete problem associated with (1.1) on the basis of the implicit Euler scheme. Let us begin by stating the required regularity assumption for the applied current source : Ω × [0, ] → R 3 , the temperature distribution : Ω × [0, ] → R, and the initial data 0 : Ω → R 3 and 0 : Ω → R 3 . Assumption 4.1. Suppose that In Assumption 4.1 and all what follows, we use the abbreviation ( ) = (·, ). This notation is also used for other functions acting in Ω × (0, ).
In the following, we shall make use of the classical discrete Gronwall lemma. For the convenience of the reader, we recall it in the following lemma: Proof. Let ∈ {1, . . . , } and ∈ {1, . . . }. The first equality of (P ) implies .

(4.8)
On the other hand, the second equality of (P ) yields .
On the other hand, from the second equality in (P ), we know that .

Combining the above two identities results in
.
Remark 5.3. (i) As pointed out in the introduction, the eddy current approximation of the Bean critical-state model leads to a parabolic quasi-variational inequality [3]. While the existence for the corresponding twodimensional case is guaranteed, no uniqueness result was derived in [3]. In the case of the original Maxwell formulation (QVI), the uniqueness is satisfied thanks to the Lipschitz property (A4). (ii) Our result extends [27,30] due to the following reasons: First, we allow for simultaneous temperature and magnetic field dependence in the critical current. Second, our result holds true for all right-hand side ∈ 1 ((0, ), 2 (Ω)) and initial data ( 0 , 0 ) ∈ 0 (curl) × (Ω) without any compatibility condition.

Further discussions
The achieved well-posedness results open a way to analyze the temperature and voltage control in the magnetization process of type-II superconductivity. This leads to a state-constrained optimal control problem governed by (QVI). This problem requires a substantial extension of the developed results [17,18,24,25,29,31] along with the recent results on the optimal control of QVI [1].