A parabolic quasi-variational inequality arising in a superconductivity model

We consider the existence of solutions for a parabolic quasilinear problem with a gradient constraint which threshold depends on the solution itself. The problem may be considered as a quasi-variational inequality and the existence of solution is shown by considering a suitable family of approximating quasilinear equations of p-Laplacian type. A priori estimates on the time derivative of the approximating solutions and on the nonlinear diffusion coefficients are used in the passage to the limit, as well as a suitable sequence of convex sets with variable gradient constraint. The asymptotic behaviour as t -~ oc is also considered, and the solutions of the quasi-variational inequality are shown to converge, at least for subsequences, to a solution of a stationary quasi-variational inequality. These results can be applied to the critical-state model of type-II superconductors in longitudinal geometry. Mathematics Subject Classification (1991): 35K85 (primary), 35K55, 35R35 (secondary). 1. Introduction In a critical-state model of type-II superconductors with a longitudinal geometry, the main unknown is the magnetic field H = (o, 0, u (x, t)), where x = (XI, x2) E S2 c JR2. By Maxwell’s equations, where E denotes the electric field, the unknown density of the induced current J is given by , In some superconductivity models (see [3]), it is assumed that the electric field E inside the superconductor depends on the current density J through This work was partially supported by the project PRAXIS/2/2.1/MAT/125/94. Pervenuto alla Redazione il 15 luglio 1999. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) Vol. XXIX (2000), pp. 153-169 154 a power law (which generalizes the classical Ohm’s law E = p J) with the scalar resistivity given by p = P(Ijl) = po1JIP-2 = where po &#x3E; 0 is a constant and p &#x3E; 2. Hence, the Maxwell’s equation for the longitudinal component of the magnetic field implies that u = u (x, t) satisfies the twodimensional quasilinear parabolic equation In Bean’s critical-state model (see [ 11 ]), the current density cannot exceed the critical value jc &#x3E; 0 and it has been suggested that this threshold may also depend on )H) I (see Kim et al. [6]). The constitutive relation for E is then modified to being ~. &#x3E; 0 an unknown Lagrange multiplier. Imposing initial and boundary conditions (in a bounded domain S2 in R 2), it may be easily shown that this problem is equivalent to the following quasivariational inequality (for details about this derivation see [II], where only the degenerate case po = 0 was considered) to a new variable h = u he, being where h e = he(t) is related to the density of external currents, as well as f ’ . This model is the main motivation for the study of this new type of quasivariational inequalities. This kind of problems seem not well studied in the literature and are not considered, for instance, in the classical references on quasi-variational inequalities [1] or [2]. Recent works in this area are, for example, the modelling works [10], [11] or a very special case of a onedimensional quasi-variational inequality considered in [7]. In Section 2 we present the main results of this paper, a result on the existence of solutions for the quasi-variational inequality and a result about the asymptotic behaviour in time of these solutions. In Section 3 we consider a family of approximating solutions and we establish apriori estimates. Section 4 includes the passage to the limit in the family of solutions of the approximated problems, concluding with the proof of the existence of at least a solution for the quasi-variational inequality in the N-dimensional case. Section 5 studies the asymptotic behaviour in time of the solutions, which, in particular, also yields the existence of a solution to the corresponding elliptic quasi-variational inequality, extending a result of [8]. 155 2. Main results Let Q be a bounded, open subset of R , with smooth boundary and let Let denote the spatial gradient and suppose that F, f’ and h are ---1 1 given functions such that where denotes the p-Laplacian, and denote the spaces of bounded measures in S2 and Q T , respectively, and We also denote by the space of Lipschitz functions that vanish on Given i we can define a convex set and the quasi-variational problem as follows: To find u in an appropriate class of functions, such that: Our first aim is to prove existence of a solution to the problem (5). For that, we consider, for 8 &#x3E; 0, fE, h, and F, smooth functions approximating, respectively, f, h and F in the norms of for any q o0 W6, (0), and in , with independent of s) and a CL function R R, non decreasing and such that We consider a sequence of the following approximated problems 156 We noticed that a similar approximation was introduced by Gerhardt (see [5]) in the treatment of the elastic-plastic torsion problem with multiply connected cross-section, which is an elliptic variational inequality and it was also used for studying a parabolic variational inequality with gradient constraint in [12]. THEOREM 1. With the assumptions (1), (2) and (3), for any 1 p 00, the problem (5) has at least a solution u belonging to T ; n C°(QT), such that ut e T; M(Q)). In addition, u is the weak limit in T ; (for any q of a of solutions of the family of approximated problems (7), u£n ~~ u in C°(QT) and also utn -1 n T ; M(Q)) weak-*. REMARK 1. Since _u is continuous and bounded, for each t e [0, T ], C C and the first integral of the left hand side of (5) should be understood in the duality sense between and taking into account that is a.e. a bounded measure. Consider now the corresponding stationary quasi-variational inequality: where we assume F satisfies (1) and Existence of a solution of the problem (8) may be proved as in [3]. We may also consider the problem (5) with T = +00. Then we have the following THEOREM 2. Suppose that the assumption (2) is satisfied for T = +00, i. e. , there exists C &#x3E; 0 such that