Applications of a version of the de Rham lemma to the existence theory of a weak solution to the Maxwell–Stokes type equation

In this paper, we show the existence of a weak solution to the Maxwell–Stokes type equation with a potential satisfying the Dirichlet condition, under the hypothesis that the domain has no holes, using a version of the de Rham lemma that was proved in our previous paper. We also give the regularity of weak solutions.


Introduction
In this paper, we consider the existence and regularity of a weak solution to a class of the Maxwell-Stokes system containing a p-curl system in a bounded domain of R 3 .
If is bounded, simply connected and has no holes, Yin [15] showed the existence of a unique weak solution for a so-called p-curl system where is the C 2,α boundary of , p > 1, n is the unit outer normal vector to and f ∈ C α ( ) is a given vector field satisfying div f = 0 in . Moreover, he got the optimal C 1,β regularity of the weak solution for some β ∈ (0, 1). See also Yin [16] and Yin et al. [17]. The system (1.1) is a steady-state approximation of Bean's critical state model for type II superconductors (cf. [9,14] Aramaki [3] extended the results to a more general system: ⎧ ⎨ ⎩ curl [S t (|curl u| 2 )curl u] = f in , div u = 0 i n , n × u = 0 on = ∂ , (1.2) where S(t) ∈ C 2 ((0, ∞)) ∩ C 0 ([0, ∞)) satisfies some structure conditions. In a multi-connected bounded domain, the systems (1.1) and (1.2) are not well posed. In this case, if has a finite connected components 0 , 1 , . . . , m with 0 denoting the boundary of the infinite connected component of R 3 \ , Aramaki [4] considered the following system: on , u · n, 1 i = 0 f o r i = 1, 2, . . . , m, ) satisfies some structure conditions given in Sect. 2 and ·, · i denotes the duality between W −1/ p, p ( i ) and Here p is the conjugate exponent of p, then we showed that (1.3) has a unique weak solution u and the weak solution belongs to C 1,β ( ), and there exists a constant C > 0 depending only on p, and f C α ( ) such that If the hypothesis div f = 0 in is not satisfied, we may consider the following Maxwell-Stokes type problem: to find (u, π) in an appropriate space such that (1.5) In this situation, the conditions corresponding to (1.4) imply (1.6) That is to say, π must be a solution of the Neumann problem for the Poisson equation. This case was considered in Aramaki [6].
In this paper, we consider the case where has no holes. If has no holes, i.e., m = 0, then the last condition of (1.5) is unnecessary. In this situation, we can consider the following Maxwell-Stokes problem: to find (u, π) in an appropriate space such that with the Dirichlet boundary condition for π instead of (1.6): In the present paper, we consider such a case. To do so, we must apply a version of the de Rham lemma. The classical de Rham lemma says that a continuous and linear functional that vanishes on all divergencefree H 1 vector fields that equal zero on the boundary can be represented as a gradient of an L 2 potential function inside the domain. For example, see Boyer and Fabrie [8,Theorem IV. 2.4]. However, the lemma does not provide any information on the trace of the potential function on the boundary.
For example, it is insufficient to show the existence of solutions for the Stokes problem. Pan [13] considered a version of the de Rham lemma. This asserts that a continuous and linear functional that vanishes on all divergence-free H 1 vector fields that have zero tangential component on the boundary is a gradient of the function π ∈ L 2 ( ), and that π has zero trace on the boundary. This additional information on the trace of π makes it possible to improve the existence of a solution of the Maxwell-Stokes type system. However, it is insufficient to consider the existence of a weak solution to the Maxwell-Stokes type problem containing a p-curl system with the potential satisfying the Dirichlet condition.
In our previous paper Aramaki [5], we considered the L p version of the result obtained by [13]. Our result is useful for the existence of a weak solution to the Maxwell-Stokes type problem in the L p version, .
The paper is organized as follows. In Sect. 2, we give some preliminaries on the trace and the gradient of functions. In Sect. 3, we give a main theorem on the existence of a weak solution to the Maxwell-Stokes type equation using the variational method under the hypothesis that has no holes. In Sect. 4, we consider the regularity of the weak solution in the case where has no holes.

Preliminaries
In this section, we state some preliminaries that are necessary later. Let be a bounded domain in R 3 with a C 2 boundary , 1 < p < ∞ and let p be the conjugate exponent i.e., , and so on, for the standard Hölder spaces, Sobolev spaces of functions. For any Banach space B, we denote B × B × B by boldface character B. Hereafter, we use this character to denote vector and vector-valued functions, and we denote the standard inner product of vectors a and b in R 3 by a · b. Moreover, for the dual space B of B, we denote the duality between B and B by ·, where u T denotes the tangential component of the trace of u, namely, u T = (n × u) × n, n is the outer unit normal vector to the boundary, and we denote its dual space by W We define the norm of elements ofĊ 1 ( ) by where γ 0 is the restriction operator to the boundary , and the completion ofĊ 1 ( ) with respect to this norm byL p ,−1/ p ( ) and define We give a proposition associated with the trace and the gradient (cf. [5]).

Proposition 2.1
Let be a bounded domain in R 3 with a C 2 boundary . Then the following holds.
(i) There exists a trace map γ : If we write and it is homeomorphic toL p ,−1/ p ( ). Now we give a L p version of the de Rham lemma that was proved by our previous paper [5].
We assume that has the following conditions as in Amrouche and Seloula [1] (cf. [2,10,12]). Let ⊂ R 3 be a bounded domain of class C r,α (r ≥ 1) with the boundary and be locally situated on one side of .

1.
has a finite number of connected components 0 , 1 , . . . , m with 0 denoting the boundary of the infinite connected component of where ∂ j denotes the boundary of j , and j is non-tangential to .
The open set˙ = \(∪ n i=1 i ) is simply connected and pseudo C 1,1 class. The number n is called the first Betti number which is equal to the number of handles of , and m is called the second Betti number which is equal to the number of holes. We say that if n = 0, is simply connected, and if m = 0, has no holes.

Existence of a weak solution to the Maxwell-Stokes type system
In this and next sections, we assume that has no holes, that is, m = dim K p N ( ) = 0. We consider the following Maxwell-Stokes type system Here u 0 T is a given vector field on such that n · u 0 T = 0 on . Moreover, f (x, z) is a given Carathéodory function on × R 3 satisfying that there exists a constant M > 0 such that Define two function spaces Definition 3. 1 We say (u, π) is a weak solution of (3.1), if (u, π) ∈ W 1, p t ( , div 0, u 0 T ) × L p ,−1/ p ( ), and satisfies π = 0 on in the sense of trace and Here if π ∈ L p ,−1/ p ( ), then ∇π ∈ W 1, p t0 ( ) is well defined from Proposition 2.1(ii). We assume that there exists a scalar function F(x, z) which is measurable in x ∈ and differentiable in z ∈ R 3 such that We are in a position to state the theorem.

Theorem 3.2 Let be a bounded domain with a C 2 boundary satisfying (1) and (2) with m = 0, that is, has no holes. Assume that a given function S(x, t) satisfies (2.3a)-(2.3d) and a given function f (x, z) satisfies (3.4) and (3.5). Then (3.1) has a weak solution
Furthermore, if f satisfies By (3.5), we can see that for any ε > 0, there exists a constant C ε > 0 such that Since has no holes, we can delete the first term u L p ( ) in the right-hand side of (3.8), that is, In fact, if (3.9) is false, there exists {v n } ⊂ W 1, p ( ) such that On the other hand, since v n W 1, p ( ) = 1, passing to a subsequence, we may assume that v n → v weakly in W 1, p ( ) and strongly in L p ( ). Hence curl v = 0, div v = 0 in and v T = 0 on . Therefore, we have v ∈ K p N ( ). Since has no holes, i.e., dim K This leads to a contradiction. Hence (3.9) holds. Therefore, we have for any u ∈ W 1, p t ( , div 0, u 0 T ). If we choose ε > 0 small enough, then it follows from (2.4) and (3.9) that there exist positive constants c, C 1 and C 2 such that Using (3.7), (3.9) and (3.10), we have If we choose ε > 0 so that εC < λ/p, we can see that {u j } is bounded in W 1, p ( ). Passing to a subsequence, we may assume that u j → u weakly in W 1, p ( ) and strongly in L p ( ). Hence div u = 0 in and u T = u 0 T on , so u ∈ W 1, p t ( , div 0, u 0 T ). Since curl u j → curl u weakly in L p ( ), it follows from Aramaki [7, Proposition 3.6] that (3.11) By the hypothesis (3.2), Summing up (3.11) and (3.12), we see that Therefore, u is a minimizer of E on W 1, p t ( , div 0, u 0 T ). For any w ∈ W 1, p t0 ( , div 0), we see that u + τ w ∈ W 1, p t ( , div 0, u 0 T ) for any τ ∈ R. Since u is a minimizer of E, we have for any w ∈ W 1, p t0 ( , div 0). By a version of the de Rham lemma (Theorem 2.2), there exists π ∈ L p ,−1/ p ( ) such that and π = 0 on in the sense of trace. Therefore, (u, π) is a weak solution of (3.1).
Furthermore, assume that f satisfies (3.6). Let (u 1 , π 1 ) and (u 2 , π 2 ) be two weak solutions of (3.1). If we take u 1 − u 2 ∈ W 1, p t0 ( ) as a test function of (3.5), we have Here we used the fact that since div u i = 0 in and π j = 0 on , by the integration by parts we have ∇π j · u i dx = 0 for i, j = 1, 2.
Thus we have If we use the monotonicity of S t (cf. [4,Lemma 3.6]): with some positive constant c, then from (3.6) we have Since has no holes, we see that u 1 = u 2 in . Taking divergence of the first equation of (3.1), π i (i = 1, 2) satisfies From the uniqueness of Dirichlet problem for the Poisson equation, we get π 1 = π 2 . This completes the proof.

Regularity of weak solutions
In this section, we consider the regularity of weak solutions to the following system: where J ∈ C α ( ) is a given function. We have the following theorem.
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