On a p-curl system arising in electromagnetism

We prove existence of solution of a $p$-curl type evolutionary system arising in electromagnetism with a power nonlinearity of order $p$, $1<p<\infty$, assuming natural tangential boundary conditions. We consider also the asymptotic behaviour in the power obtaining, when $p$ tends to infinity, a variational inequality with a curl constraint. We also discuss the existence, uniqueness and continuous dependence on the data of the solutions to general variational inequalities with curl constraints dependent on time, as well as the asymptotic stabilization in time towards the stationary solution with and without constraint.


Introduction
We consider a nonlinear electromagnetic field in a bounded domain Ω of R 3 . The electric and the magnetic fields, respectively e = e(x, t) and h = h(x, t), and the electric and magnetic inductions, respectively d(x, t) and b = b(x, t), satisfy the Maxwell's equations (∂ t = ∂ ∂t , ∇× = curl, ∇· = div) where j denotes the total current density, q is the electric charge and f , which is zero in the classical setting, is here a given internal magnetic current (see [3, 2000 Mathematics Subject Classification. Primary: 35K87, 78M30; Secondary: 49J40. Key words and phrases. Electromagnetic problems; variational methods; variational inequalities; superconductivity models. The first an last authors are supported by the Research Centre of Mathematics of the University of Minho through the FCT Pluriannual Funding Program and FCT project UT-Austin/MAT/0035/2008. 6]). Denoting by µ the magnetic permeability constant, we assume the following constitutive law b = µh and the following nonlinear extension of Ohm's law, where σ is the electric conductivity.
If in the first equation of (1) we neglect the term ∂ t d, the magnetic field h is then divergence free and µ ∂ t h + ∇× 1 σ |∇×h| p−2 ∇×h = f . Denoting Γ = ∂Ω and Σ T = Γ×(0, T ), we impose the following natural tangential boundary conditions h · n = 0 and e × n = g on Σ T , where n denotes the external unitary normal vector to the boundary Γ. The boundary condition h · n = 0 is naturally associated with ∇ · h = 0 in Q T = Ω × (0, T ) and e × n = g corresponds to consider a superconductive wall, i.e., a tangent current field.
Recalling the relation between e and h, if we set ν = 1 σ > 0, we are lead to the problem h · n = 0 and ν|∇×h| p−2 (∇×h) × n = g on Σ T , As a necessary condition for the existence of solution of this problem, the external field f must satisfy ∇·f = 0. Besides, the given field g on Σ T must be tangential and compatible with f , more precisely, ∇ Γ · g = f · n on Γ, where ∇ Γ · denotes the surface divergent (see [9,10,8]).
We may also consider another constitutive law that arises in type-II superconductors and is known as an extension of the Bean critical-state model presented in [11]. In this case the current density cannot exceed the critical value Ψ > 0 and we have where the parameters ν = ν(x) ≥ 0 is a given function and λ = λ(x, t) ≥ 0 can be regarded as a (unknown) Lagrange multiplier. Some easy calculations (see [11,8] for details) leads to the variational inequality, for a.e. t ∈ (0, T ), for any test function v = v(x) such that |∇×v(x)| ≤ Ψ(x, t). This leads to search the solution in the time dependent convex set K(t) = {v = v(x) : |∇×v(x)| ≤ Ψ(x, t), x ∈ Ω} for a.e. t ∈ (0, T ).
In Section 2 we study the evolutionary problem (2), showing the existence of a unique solution in the variational framework of quasilinear monotone operators in the appropriate functional subspace of W 1,p (Ω) 3 . We notice that in the case of normal boundary condition (h×n = 0 on Σ T ) existence results for similar nonlinear Maxwell's system have been obtained in [18,19]. But these results with tangential boundary condition (h · n = 0 on Σ T ) are presented here for the first time. We also prove the asymptotic convergence, as t → ∞ to the stationary solution of the problem already considered in [8].
In Section 3 we derive the Bean-type superconductivity variational inequality model with critical value Ψ = 1 as the limit case p → ∞, extending a previous scalar case by [2] and a vectorial case with normal boundary condition due to [19].
Finally, in Section 4, we solve the evolutionary variational inequality (3) with the time dependent convex set (4), showing the existence, uniqueness and continuous dependence on the data f , g, h 0 and Ψ of the solution, in the appropriate setting. We also discuss the asymptotic convergence of the solution in L 2 (Ω), as t → ∞, towards the corresponding stationary solution obtained in [8], for p ≥ 6 5 .

The variational equation
In what follows Ω is a bounded, simply connected domain of R 3 with a C 1,1 boundary Γ. If E denotes a vectorial space, we denote by E the space E 3 .
2.1. The functional framework. We introduce the functional space Remark 1. Two immediate consequences follow from this proposition: there exist positive constants C q and C r such that, given v ∈ W p (Ω), the Sobolev inequality holds with q ≤ 3p 3−p if 1 < p < 3, any q < ∞ if p = 3 and q = ∞ for p > 3 and the holds with r ≤ 2p 3−p if 1 < p < 3, any r < ∞ if p = 3 and r = ∞ for p > 3. In particular, v ∈ L 2 (Ω) if p ≥ 6 5 . In what follows the exponents p, q and r are related by these Sobolev and trace inequalities.

2.2.
Existence of solution in the evolution problem. Let a : Q T × R 3 −→ R 3 be a Carathéodory function satisfying the structural conditions for given constants a * , a * > 0, for all u, v ∈ R 3 and a.e. (x, t) ∈ Q T . We consider the following problem: h · n = 0 and a(x, t, ∇×h) × n = g on Σ T , Taking (5) and (6) into account we assume that where q ′ and r ′ denote the conjugate exponents of q and r respectively, and Hence the following formula of integration by parts holds with a ∈ L p ′ (Ω), ∇×a ∈ L q ′ (Ω) and, in the sense of traces, a × n |Γ ∈ L r ′ (Γ) (see [4] and [9]).
Whenever ∂ t h(t) ∈ W p (Ω) ′ , interpreting the integral Ω ∂ t h · ϕ in the duality sense, the above formula yields the following weak formulation of the problem (8): to find h ∈ L p (0, T ; W p (Ω)) such that, for a.e. t ∈ (0, T ), Proposition 2. Suppose that the operator a satisfies the assumptions (7a-c) and the data and the initial condition satisfy (9) and (10). Then the problem (8) has a unique solution h ∈ L p (0, T ; W p (Ω)) ∩ C(0, T ; L 2 σ (Ω)) and ∂ t h ∈ L p ′ (0, T ; W p (Ω) ′ ). In addition, there exists a positive constant C such that Proof. The operator A(t) : W p (Ω) −→ W p (Ω) ′ defined for a.e. t ∈ (0, T ) by is a uniformly bounded (independently of t), hemicontinuous, monotone and coercive operator, due to the structural properties (7a-c). Defining, for a.e. t ∈ (0, T ), and adapting a well-known existence theorem to monotone operators independent of t (see [7]), we easily prove that problem (12) has a solution in L p (0, T ; W p (Ω)) ∩ C(0, T ; L 2 σ (Ω)). The uniqueness of solution results directly from the strict monotonicity (7c) of the operator A.
To obtain the estimate (13) choose h(t) as test function in (12). Denoting Q t = Ω × (0, t) and Σ t = Γ × (0, t), we have Applying Hölder and Young inequalities and the Remark 1, we obtain and the conclusion follows.
Remark 2. The functional framework we introduced provides a general variational setting for the stationary solutions of (8). Indeed, for instance for arbitrary f ∈ L q ′ (Ω), g ∈ L r ′ (Γ) and ν ∈ L ∞ (Ω), ν(x) ≥ a * > 0 for a.e. x ∈ Ω, the unique minimum of the functional in W p (Ω), provides the weak stationary solution to (8). However, as remarked in [8] in the stationary problem, for the existence of solution of the strong boundary value problem (8) with given data (f , g), it is necessary that f is divergence free and g is tangential and compatible with f (∇ Γ · g = f · n) on Γ. But the weak formulation (12) of the problem (8) has a unique solution with no restrictions on the data.
Usually, a weak equation is also a strong one, as long as it has enough regularity. The situation here requires also additional compatibility conditions, since we are working with strongly coupled systems and the test functions have strong restrictions (they are divergence free and tangential on the boundary). Indeed, given f ∈ L q ′ (Ω), the Helmholtz decomposition (see [14]) gives us that f = f 0 + ∇ξ, where f 0 is divergence free. On the other hand, if g ∈ L r ′ (Γ), g = g T + g N , where g T and g N are, respectively, the tangential and the normal components of g. So, the set of test functions W p (Ω) only takes into account f 0 (the divergence free component of f ) and g T (the tangential component of g) and consequently the problems (12) with data (f , g) and (f 0 , g T ) yields the same solution and both correspond to the weak formulation of the problem (8) with data (f 0 , g T ).
In the particular case where we can improve the Proposition 2 assuming more regularity on the data.
Proof. Using Galerkin approximations (see for instance [7] or Chapter 3 of [20]), we may set formally ∂ t h(t) as test function in (12). Integrating between 0 and t leads to Noting that and so 2.3. The asymptotic behaviour when t → ∞. In this section we give sufficient conditions in order to establish that where h denotes the solution of the problem (8) and h ∞ solves the stationary problem by applying the integration by parts (11). and Theorem 2.1. Let p > 2, suppose that the operators a and a ∞ satisfy (7 a, b, c') and Proof. Choosing for test function in (18), for a.
Taking w(t) as test function in (12), for a.e. t ∈ R + , we conclude that Since, by (7c'), subtracting (20) from (21), using Hölder and Young inequalities and the Remark 1, we have the inequality (22) is written as follows Theorem 2.3. Let p = 2 and suppose that the operators a and a ∞ verify (7 a, b, c') and Proof. Arguing as in the previous theorem, calling from which we obtain, using Hölder and Young inequalities and the Remark 1, and l 0 is a constant which exists by the assumptions on a, a ∞ , f , f ∞ , g and g ∞ .
In order to prove that w ∈ L ∞ (0, ∞; L 2 (Ω)), we multiply (23) by e Ct and integrate in time, between σ and τ , σ ≤ τ . Then Combining (24) and (25) we get and taking τ = t and σ = 0, there exists a positive constant l 1 such that, for all t, Lemma 2.4 ([5], p 286). Let φ(t) be a nonnegative function, absolutely continuous in any compact interval of R + , l(t) a nonnegative function belonging to L 1 loc (R + ) and c a positive function such that Then we have Proof. By the property (7 c'), Setting w(t) = h(t) − h ∞ , recalling (21) and using the above inequality, we obtain We recall now the inverse Hölder inequality (see [17], p 8): let 0 < s < 1 and and From (18) and the assumptions we have Simple calculations allows us to rewrite the inequality (17) in the form We get, using the Proposition 3, and, from (26), By the Remark 1 we know, since p ≥ 6 5 , that 1 2 and so, for C = 2C 5 and l(t) = 2D 1 ξ(t) we deduce that and the proof is concluded exactly as the previous one.
Proof. Choosing h n as test function in (31) and using the Remark 1, we obtain where C 1 is a positive constant independent of n.
On the other hand, formally we have from (31), with ϕ = ∂ t h n (t), and, on the other hand, we have for any fixed 3 < q < ∞, where h * is the solution of the problem (32).
Proof. By the uniform estimates in (33) we only need to check that h * solves (32). Let ϕ ∈ W p (Ω) be such that |∇×ϕ| < 1 a.e.. Taking ϕ − h n as a test function in (31), we have By the monotonicity of the operator a n defined in (30) we have Applying limit in n to both members of (36) we get Since ϕ is an arbitrary function of W p (Ω) satisfying |∇×ϕ| < 1, the inequality (37) still holds, by density, for all ϕ ∈ K * . We also have h * (0) = h 0 .
Remark 5. If we apply Theorem 2.1 for each fixed n > 3 with where the constant C > 0 is independent of n and t.
So, for a subsequence n ′ satisfying (38), there exists a sequence, t n ′ → ∞, such that An interesting open question in the degenerate case is whether there exists a sequence t n → ∞ such that h * (t n ) converges, in some sense, to h * ∞ .

The variational inequality with evolutionary curl constraint
Define, for a.e. t ∈ (0, T ), the following closed convex subset of W p (Ω), where Ψ : Q T −→ R + is a function such that Ψ ≥ α > 0.
We define the variational inequality: to find h, in a suitable class of functions, such that ∀ϕ ∈ K(t), for a.e. t ∈ (0, T ).

4.2.
Existence of solution of the variational inequality. In order to prove that a subsequence of the solutions of the approximate problems converges, with ǫ → 0, for the solution of the variational inequality, we need additional a priori estimates.
Choosing in (41) for test function ϕ − h ε (t), being ϕ ∈ K(t) and integrating in time, we obtain Assuming that h(t) ∈ K(t) for a.e. t ∈ (0, T ) (this fact will be proved in the next lemma), applying a variant of Minty's Lemma and standard arguments, we conclude where ϕ is any function belonging to K(t), for a.e. t ∈ (0, T ).
The uniqueness is immediate.

Proof. Define
we conclude that |∇×h| ≤ Ψ a.e. in Q T , completing the proof.
and so h 2 (t) ∈ K 2 (t) for a.e. t ∈ (0, T ). Now, Integrating in time we obtain Remark 6. If we replace, in the last lemma, the subscript 1 by the subscript 2, the corresponding function we construct will be denoted by h 1 .
As in Lemma 4.6 we have h(t) ∈ K ∞ and h ∞ (t) ∈ K(t), for a.e. t ∈ (0, ∞). Substituting, in (51), h 1 by h and h 2 by h ∞ , we obtain and h(t) and h ∞ (t) are defined in (53). From Lemma 4.3, we observe that there exists positive constants, C 1 and C 2 , independent of t, such that ∂ t h L 2 (Ω×(0,t)) ≤ C 1 t Arguing as in Section 2.3, for p > 2, we obtain, for a positive constant C, where, for a positive constant C 3 , where, for a positive constant C 3 , because γ > 1 2 . Arguing, in both cases, exactly as in the Section 2.3, the conclusion follows.