A Class of Variational–Hemivariational Inequalities for Bingham Type Fluids

In this paper we investigate a new class of elliptic variational–hemivariational inequalities without the relaxed monotonicity condition of the generalized subgradient. The inequality describes the mathematical model of the steady state flow of incompressible fluid of Bingham type in a bounded domain. The boundary condition represents a generalization of the no leak condition, and a multivalued and nonmonotone version of a nonlinear Navier–Fujita frictional slip condition. The analysis provides results on existence of solution to a variational–hemivariational inequality, continuous dependence of the solution on the data, existence of solutions to optimal control problems, and the dependence of the solution on the yield limit. The proofs profit from results of nonsmooth analysis and the theory of multivalued pseudomontone operators.

hand, due to nonconvexity of the potential, the problem can not be treated within the theory of variational inequalities, as in [2]. To the best of our knowledge, the Bingham model has not been studied so far by the theory of hemivariational inequaties. The weak formulation of the problem studied in the paper naturally leads to a variationalhemivariational inequality.
The third novelty is to establish continuous dependence of solutions on the second member and demonstrate the existence of solutions to an optimal control problem for the Bingham model. Further, prove a new convergence result which, in particular, shows that if the yield stress tends to zero, then the Bingham fluid tends to behave as a Newtonian one.
Note that the frictional boundary condition is described by a nonlinearity of the form k∂ j. In general, this kind of nonlinearity can not be represented by a purely hemivariational inequality since there is not a potential G with ∂G = k∂ j. This type of nonlinearity leads to a "hemivariational term" on the boundary in the inequality while the Bingham law leads to a "variational part" in the domain. Various models in fluid mechanics have studied within the theory of variational-hemivariational inequalities in [25] for Navier-Stokes equations, in [11,12] for non-Newtonian and generalized Newtonian fluids, in [13] for stationary and evolutionary Oseen models.
The rest of the paper is structured as follows. After a preliminaries in Sect. 2, in Sect. 3 we deliver both classical and variational formulation of the Bingham model, and state the main existence result of the paper, Theorem 4. Section 4 is devoted to the proof of Theorem 4. In Sect. 5 we provide a result on the continuous dependence of solutions on the second member, analyse optimal control problem for the variationalhemivariational inequality, and study the dependence of the solution on the yield limit. Finally, Sect. 6 provides a short indication to open problems for the future work.

Mathematical Background
The study on variational inequalities uses results from nonsmooth analysis and the theory of monotone operators which comprehensive presentation can be found in [7,9,10,26,28].
Let X be a normed space with norm denoted by · X . Let X * stand for its topological dual, and the duality brackets for the pair (X * , X ) is denoted by ·, · X * ×X . If no confusion arises, we often skip the subscripts. The weak and norm convergences in X are denoted by " " and "→", respectively. By L(E, F) we denote the space of linear and bounded operators from a normed space E to a normed space F endowed with the operator norm · L(E,F) . Given a set G ⊂ X , we set G X = sup{ x X | x ∈ G}.
Single-valued monotone operators Let X be a Banach space. An operator A : X → X * is said to be monotone, if for all u, v ∈ X , it holds Au− Av, u−v ≥ 0. It is called maximal monotone, if it is monotone, and Au −w, u −v ≥ 0 for any u ∈ X entails w = Av. Operator A is bounded, if A maps bounded sets of X into bounded sets of X * . Operator A is called pseudomonotone, if it is bounded and u n u in X with lim sup Au n , u n − u ≤ 0 imply Au, u − v ≤ lim inf Au n , u n − v for all v ∈ X . It is known that if X is a reflexive Banach space, then A : X → X * is pseudomonotone, if and only if it is bounded and u n u in X with lim sup Au n , u n − u ≤ 0 imply lim Au n , u n − u = 0 and Au n Au in X * .
Multivalued monotone operators For an operator T : X → 2 X * , its domain D(T ) and graph Gr(T ) are defined, respectively, by An operator T : X → 2 X * is called monotone when u * − v * , u − v ≥ 0 for all (u, u * ), (v, v * ) ∈ Gr(T ). It is said to be maximal monotone, if it is monotone and maximal in the sense of inclusion of graphs in the family of monotone operators from Let X be a reflexive Banach space. An operator T : X → 2 X * is pseudomonotone if: (a) for every u ∈ X , the set T u ⊂ X * is nonempty, closed and convex, (b) T is upper semicontinuous from each finite dimensional subspace of X into X * endowed with weak topology, (c) for all sequences {u n } ⊂ X and {u * n } ⊂ X * such that u n u in X , u * n ∈ T u n for all n ≥ 1 and lim sup u * n , u n − u ≤ 0, we have that for every v ∈ X , there exists Let X be a reflexive Banach space. An operator T : X → 2 X * is generalized pseudomonotone if for any sequences {u n } ⊂ X and {u * n } ⊂ X * such that u n u in X , u * n ∈ T u n for n ≥ 1, u * n u * in X * and lim sup u * n , u n −u ≤ 0, we have u * ∈ T u and lim u * n , u n = u * , u . It is known that if T is pseudomonotone, then it is generalized pseudomonotone. Conversely, if T is a bounded generalized pseudomonotone operator such that for all u ∈ X , T u is a nonempty, closed and convex subset of X * , then T is pseudomonotone.
Generalized subgradient Let X be a Banach space. Let ϕ : X → R ∪ {+∞} be a proper, convex and lower semicontinuous function. The mapping ∂ϕ : X → 2 X * defined by is called the subdifferential of ϕ. An element x * ∈ ∂ϕ(x) is called a subgradient of ϕ at x. Let h : X → R be a locally Lipschitz function. The generalized (Clarke) directional derivative of h at the point x ∈ X in the direction v ∈ X is defined by The generalized subgradient of h at x is a subset of the dual space X * given by

Statement of the Flow Problem
In this section we provide the classical and variational formulations of the flow model of a Bingham In Problem 1, the flow of an incompressible fluid is governed by the conservation law (1). The tensorial constitutive law (2) is a relation between the extra stress tensor S and the strain velocity tensor Du whose components are given by The law (2) represents the Bingham model which is used to analyze flows of materials for which the imposed stress must exceed a critical yield stress (called plasticity threshold) g to initiate motion, i.e., they behave as rigid bodies when the stress is low but flow as viscous fluids at high stress. Examples of Bingham fluids include cosmetics and personal care products, water suspensions of clay, drilling muds, volcanic lava and magmas, or sewage sludge. For g = 0 and μ(r ) = μ 0 , the constitutive law reduces to the model of Newtonian fluid for the Navier-Stokes equation. Here, μ : [0, ∞) → R is a given positive viscosity function, g : → [0, +∞) stands for the yield limit, and ρ is the constant density of the fluid, conveniently put equal to one. The solenoidal (divergence free) condition (3) of the form div u = ∇ · u = d i=1 D ii (u) = 0 in states that the fluid is incompressible. The homogeneous Dirichlet boundary condition (4) means that the fluid adheres to the wall, see [26].
The symbols Div and div denote the divergence operators for tensor and vector valued functions S : × M d → M d and u : → R d defined by Div S = (S i j, j ) and div u = (u i,i ), and the index that follows a comma represents the partial derivative with respect to the corresponding component of x. The tensor product of two vector fields v, w ∈ R d is the second-order tensor σ defined by The contracted product of tensors σ and τ of order 2 is the real number defined by Alternatively, the contracted tensor product is simply the Euclidean scalar product in the space of the matrices identified with R d×d . For a vector-valued function v on , we denote by v ν and v τ its normal and tangential components defined by v ν = v · ν and v τ = v − v ν ν, respectively. Further, the total stress tensor is expressed by where I is the identity d × d matrix. The traction vector is denoted by τ (u, p) = σ (u, p)ν on , so τ ν (u, p) = τ (u, p) · ν and τ τ (u) = τ (u, p) − τ ν (u, p)ν represent normal and tangential components of the traction vector, respectively. Note that τ and τ ν do depend on the pressure p, and τ τ is independent of p. To simplify the notation, we often do not indicate explicitly the dependence of various functions on the spatial variable x ∈ ∪ . The inner products and norms on R d and M d are denoted by the intuitive notation.
For the analysis of the problem, we need the following spaces The space V is endowed with the standard norm v = v H 1 ( ;R d ) . We also consider the norm given by Using the Korn inequality, see, e.g., [11,Theorem 8], it is known that · H 1 ( ;R d ) and · V are the equivalent norms on V . By ·, · we denote the duality brackets for the pair is continuous and compact. We denote its norm in the space L(V , L 2 ( ; R d )) by γ . Moreover, instead of γ v, for simplicity, we often write v.
We need the following hypotheses on the data.
In hypothesis H ( j) (iii), and in what follows, ∂ j denotes the generalized gradient of the function j with respect to its last variable.
We shall explain the boundary condition (5). The first condition in (5) is called the impermeability (no leak) boundary condition, and the second one is called the nonmonotone slip boundary condition.
(ii) The nonlinear Navier-type slip condition where h : is prescribed, was used in [21] to model the wall slip of non-Newtonian fluids.
(iii) The following threshold slip condition or the Navier-Fujita condition of frictional type where α : 1 → (0, ∞) is a given prescribed slip threshold, has been treated in [15-17, 32, 33]. Condition (13) can be formulated as follows The version of (13) of the form where w τ denotes the tangential velocity of the wall surface 1 can be found in [20,22]. (iv) The nonlinear Navier-Fujita slip condition where α : 1 → (0, ∞) and h : 1 × [0, ∞) → [0, ∞) are given bounded functions such that for a.e. x ∈ 1 , h(x, r ) = 0 if and only if r = 0, has been studied in [21,22]. This condition is a generalization of Navier's slip condition and a Coulomb-type friction condition. Condition (14) says that the fluid slips at the boundary points if the magnitude of the tangential traction reaches a critical value α being a prescribed threshold on the boundary wall, and when the slip occurs, the tangential traction equals to a given nonlinear function of the slip velocity. We observe that the Navier-Fujita law of the form where where ∂ j represents the subdifferential of the convex function j : at the points on 1 where u τ = w τ , and at the points on 1 which implies (16).
(v) Note that condition (5) is much more general than the Navier-Fujita slip condition (15) and leads to novel models. It allows to model nonmonotone slip boundary conditions of frictional type governed by nonconvex locally Lipschitz potentials. For example, let j : R d → R be defined by where θ ≥ 0 is a given constant. For simplicity, here we take w τ = 0 on 1 . The function j is nonconvex for θ > 0 and its generalized gradient is given by where B(0, 1) is the closed unit ball in R d . This choice of j leads to the slip condition of the form This condition illustrates the slip weakening phenomenon in which the tangential traction is a decreasing function of the tangential velocity. It is clear that for θ = 0 we get (15) with k(ξ ) = F( ξ ) and that j satisfies H ( j) with c 0 (x) = 1 + θ and c 1 = 0, so for this potential the smallness condition (25) below holds trivially. It is obvious that we can deal with a slip law (17) in which θ ∈ L ∞ ( 1 ), θ ≥ 0 a.e. on 1 . This allows for varying the slip condition on different parts of the boundary 1 .
Example 2 Let d = 2 and τ be the (only one) tangential direction on the boundary. Then τ τ = τ τ τ and u τ = u τ τ , where τ τ and u τ are real-valued functions. Consider the nonconvex locally Lipschitz function j : R → R given by Then the law (5) reduces to the nonmonotone slip condition where sgn(r ) = 1 if r > 0, and −1 if r < 0. Thus, on [−1, 1] the slip condition is represented by a monotone graph, and for u τ < −1 and u τ > 1 the tangential traction is a decreasing function of the tangential velocity.
To provide the variational formulation, we suppose that u, S and p are sufficiently smooth functions that satisfy Problem 1. Let v ∈ V be a test function. We multiply (1) by v − u and apply the Green-type formula, see [26,Theorems 2.25] to obtain Since div v = div u = 0 in and v ν = u ν = 0 on , by [26, Theorem 2.24], we easily get Using the above equality and the properties u = v = 0 on 0 and (u ⊗ u)ν = 0 on 1 (19) in (18), we have From the condition (5), H (k)(iii) and the definition of the subgradient, it follows that Combining the well known decomposition formula, see [26,(6.33)] and the property Let + = {x ∈ | Du(x) > 0} and 0 = {x ∈ | Du(x) = 0}. Exploiting the constitutive law (2), by the Cauchy-Schwarz inequality, we get On the other hand, by (2) Adding the inequalities (22) and (23), we deduce Finally, we use (21) and (24) in (20) and obtain the following variational formulation of Problem 1.

Theorem 4 Under the hypotheses H (μ), H( f , g), H( j) and H (k), and the smallness condition
Problem 3 has a solution.
The inequality in Problem 3 is a variational-hemivariational inequality. It combines the convex potential v → g Dv dx and the locally Lipschitz, in general nonconvex, superpotential j. When j(x, ·) is a convex function for a.e. x ∈ 1 , then from Problem 3 we obtain a purely elliptic variational inequality of second kind: find a velocity u ∈ V such that

Proof of Theorem 4
The proof is based on an application of a surjectivity result for pseudomonotone and coercive operators, see [10,Theorem 1.3.70]. We divide the proof into several steps.
Step 1 Let X = L 2 ( 1 ; R d ). We introduce the following operators and functions Under the above notation, the variational-hemivariational inequality in Problem 3 can be equivalently written as follows: find u ∈ V such that for all v ∈ V . In order to show existence of solution to (31), we consider the auxiliary inclusion problem.

PROBLEM 5 Find u ∈ V such that
In Problem 5, the notation ∂ϕ denotes the subdifferential in the sense of convex analysis of ϕ, and M * : X * → V * is the adjoint operator to M. The notation ∂ J stands for the generalized subgradient of the function J (w, ·). Under hypothesis H ( j), we know, see [26,Theorem 3.47(i)-(iv)], that J is well defined on X × X , J (w, ·) is Lipschitz on bounded subsets of X for all w ∈ X (and hence also locally Lipschitz on X ), and the following inequality holds We observe that every solution to Problem 5 is also a solution to problem (31). Indeed, let u ∈ V be a solution to Problem 5, which means that where ξ ∈ ∂ J (M u, M u) and η ∈ ∂ϕ(u). Hence, by the definitions of subgradients, we get Let v ∈ V . Then, by (33), we have We combine (35) with (34) and (32) to deduce (31). We conclude that u ∈ V is a solution to the inequality (31).
Step 2 We will show that Problem 5 has a solution. To this end, we use the well known result from the theory of monotone operators, see [10, Theorem 1. 3.70] which states that if X is a reflexive Banach space and T : X → 2 X * is a multivalued pseudomonotone and coercive operator, then T is surjective. Let We will verify the following properties.
• The operator A : V → V * given by (26) is bounded, monotone and continuous. We use H (μ)(i) and the Hölder inequality to get The latter implies that A maps bounded sets in V into the bounded sets in V * , i.e., A is a bounded operator. Next, from hypothesis H (μ)(ii), we directly have Hence, we have We use the fact that the embedding operator V ⊂ L 4 ( ; R d ) is continuous and compact for d = 2, 3, see (8). This fact and the generalized Hölder inequality imply that the integral in B is well defined as a product of two functions in L 4 ( ; R d ) and a function in L 2 ( ; R d ). The boundedness of the operator B follows from the estimate Subsequently, we prove that B is completely continuous, that is, it maps weakly convergent sequences in V to strongly convergent ones in V * . Let {u k } ⊂ V be such that u k u in V , as k → ∞. Therefore, we have u k → u in L 4 ( ; R d ). From the estimate Let v ∈ V . We apply the Hölder inequality to obtain with a constant c > 0 independent of k. Hence, by (36), Bu k − Bu V * → 0, as k → ∞ which proves that B is the completely continuous operator. Using, for instance, the characterization of pseudomonotonicity of [26, Proposition 3.66], we know that every bounded and completely continuous map is pseudomonotone. We deduce that B : V → V * is a pseudomonotone operator.
• Consider now the multivalued operator T 1 : V → 2 V * defined by We will prove that T 1 is a bounded, multivalued pseudomonotone operator. First, we note that by hypotheses H ( j), H (k), and [26, Theorem 3.47(v), (vi)], we obtain where c 3 , c 4 ≥ 0. In particular, this estimate shows that the operator ∂ J : X × X → 2 X * is bounded, recall that the subdifferential is taken with respect to the last variable of a function. Second, we show that T 1 is a bounded operator. Let D be a bounded subset of V . Since A + B and M are bounded operators, it is clear that (A + B)v and Mv for v ∈ D are bounded in V * and X , respectively. The boundedness of ∂ J implies that ∂ J (Mv, Mv) for v ∈ D remains in a bounded subset of X * , and subsequently M * ∂ J (Mv, Mv) for v ∈ D is bounded in V * . Thus T 1 is a bounded operator as a sum of bounded operators.
Third, we note that T 1 v is a nonempty, closed and convex subset of V * for all v ∈ V . Indeed, we know, see [26,Proposition 2.23(iv)], that for all w ∈ X , the generalized gradient ∂ J (w, w) is a nonempty, convex and weakly compact set in X * . Hence, this set is also weakly closed, and by convexity, it is closed in X * . Therefore, the set M * ∂ J (Mv, Mv) is nonempty, closed and convex for all v ∈ V , and the values of T 1 are nonempty, closed and convex too.
Fourth, in order to prove that T 1 is pseudomonotone, by Proposition [10, Proposition 1.3.66], it is enough to show that T 1 is generalized pseudomonotone. To this Mv n ). It is obvious that {v n } is bounded in V and {Mv n } is bounded in X . Using the boundedness of ∂ J , we obtain that {w n } remains in a bounded subset of X * . Hence, by passing to a subsequence, if necessary, we may assume that w n w in X * with w ∈ X * . (38) On the other hand, by the compactness of operator M, see (9), we infer that Exploiting (38) Finally, we are able to pass to the limit in the equality v * This completes the proof that the operator T 1 is pseudomonotone.
• Consider now the multivalued operator T 2 : V → 2 V * defined by with ϕ : V → R given by (29). We will prove that T 2 is a bounded, multivalued pseudomonotone operator. Let I : L 2 ( ; M d ) → R be the functional defined by Note that, by H ( f , g) We apply [3,Proposition 2.53] to deduce that the functional I is convex, lsc, everywhere finite, and ∂h(x, w(x)) a.e. x ∈ for all w ∈ L 2 ( ; M d ). The latter implies that is convex, lsc, and finite.
is linear and continuous, we use the chain rule to obtain From [10,Theorem 1.3.19] and (40), it follows that ∂ϕ : V → 2 V * is a maximal monotone and bounded operator, and D(∂ϕ) = V . In consequence, by [10, Corollary 1.3.67], the operator T 2 : V → 2 V * is bounded and pseudomonotone.
Let u ∈ V . From hypothesis H (μ)(i), we have Next, we use the integration by parts formula, see [26,Theorem 2.23], and the conditions div u = 0 in , u = 0 on 0 , and u ν = 0 on 1 to get Further, we profit from H ( j)(iii), H (k)(iii) and (32), to deduce that for every ξ ∈ ∂ J (M u, M u), it holds Recalling that the trace operator γ : V → X is continuous, we arrive at Also, for the convex subdifferential term, we use (40) to obtain Combining (41), (42), (43) and (44), we have Taking into account the smallness condition (25), we deduce that T : V → 2 V * is a coercive operator.
In conclusion, the operator T : V → 2 V * is pseudomonotone and coercive, and so it is surjective. Therefore, for each f ∈ V * , Problem 5 has a solution. The proof is now complete.
The existence results for Problem 3 in Theorem 4 extends [24, Theorem 4.1] obtained for the Oseen model under more restrictive (Clarke regularity) hypothesis on j(x, ·) and on the viscosity function μ. Further, we have also neglected the hypothesis on the relaxed monotonicity of ∂ j(x, ·) for a.e. x ∈ 1 which has been widely used for hemivariational inequalites, see [26,27] and the references therein.
We state a version of Theorem 4 which provides a sufficient condition for existence of solution without the smallness condition (25).
Proof Under the growth condition (45), the coercivity condition of the operator T reads as follows for all u ∈ V . The rest of the proof is similar to the one of Theorem 4.
We complete this section by showing a possible passage from Problem 3 to Problem 1 in the case of regular solutions and g = 0.
Proposition 7 Let u ∈ H 2 ( ; R d ) be the solution to Problem 3 with g = 0. Then there is p ∈ L 2 ( ) with p dx = 0 such that (u, p) satisfies conditions of Problem 1.
Proof Let u ∈ V be a smooth solution to Problem 3 for g = 0. We recover the pressure in Problem 3. Let ψ ∈ C ∞ 0 ( ; R d ) be such that div ψ = 0 in . Define v := u ± ψ. Then v ∈ V , so we choose it as a test function in Problem 3 to get This means that the linear functional L : We are in a position to apply [35,Lemma 2.2.2,p.75] to deduce that that there exists a unique p ∈ L p ( ) such that p dx = 0 and L = ∇ p in the sense of distributions. By the definition of the divergence operator, we have Since ψ is arbitrary, by the variational lemma, see [37,Proposition 18.6], we deduce which proves that (1) with condition (2) holds (recall that, for convenience, we have assumed that ρ ≡ 1). Subsequently, we interpret the boundary condition (5). We multiply the equation (46) by a suitable test function and compare the result with the inequality in Problem 3. Let ψ ∈ C ∞ ( ; R d ) with div ψ = 0 in , ψ = 0 on 0 and ψ ν = 0 on 1 . We choose v := u + ψ, we use the properties of ψ and the fact u ∈ V to infer that v ∈ V . We multiply (46) by v − u and get Analogously as in the weak formulation (see Sect. 3), we have and Inserting (48) and (49) into (47), we obtain On the other hand, Problem 3 for v = u + ψ implies Combining (50) with (51), we get Note that Div(u⊗u)·ψ dx = − (u⊗u) : Dψ dx. Since ψ is arbitrary, choosing We use the definition of the generalized gradient to have −τ τ (u) ∈ k(u τ )∂ j(u τ ) on

Optimal Control Problem and Convergence Result
The purpose of this section is twofold. First, we analyse the class of optimal control problems for a system governed by the variational-hemivariational inequality in Problem 3. Second, we study the dependence of the solution on the yield limit g.
Let U = H = L 2 ( ; R d ) be the space of controls representing external body forces. For every f ∈ U , let S( f ) ⊂ V be the solution set to Problem 3 corresponding to f . From the previous section, we know that under the hypotheses of Theorem 4 or of Theorem 6, the set S( f ) is nonempty. Consider the following optimal control problem. Given a nonempty subset U ad of U denoting the set of admissible controls, and an objective functional F : U × V → R, F = F( f , u), find f * ∈ U ad and a state u * ∈ S( f * ) such that (52) A pair which solves (52) is called an optimal solution. We will show the existence of optimal solutions to (52) using a result on the continuous dependence of solutions on the right hand side. The proof under the hypotheses of Theorem 6 is analogous. We infer that there exists u n ∈ V such that (see (31)) for all v ∈ V , where A, B and ϕ are defined by (26), (27) and (29), respectively. First we show the a priori bound for the solutions. To this end, we take v = 0 as a test function in (53) to obtain From (41) and (42), we know that (A + B)u n , u n ≥ μ 0 u n 2 V . By H ( j)(iii), H (k)(iii), the continuity of γ , see (9), and [26, Proposition 3.23(iii)], we have We also get ϕ(0) = 0 and ϕ(u n ) ≥ 0. By the properties stated above, it follows that and next by the smallness condition (25), we deduce From (54) we deduce that {u n } is bounded in V uniformly with respect to n. Hence, by passing to a subsequence if necessary, we may assume that there exists u ∈ V such that It remains to show that u ∈ S( f ). From the compactness of the trace operator γ , see (9), and [26, Theorem 2.39], we may suppose that at least for a next subsequence, we have u n → u in L 2 ( 1 ; R d ) and Let v = u be a test function in (53). We have We use [26, Proposition 3.23(ii)] and convergence (56) to get lim sup j 0 (u nτ ; u nτ − u τ ) ≤ j 0 (u nτ ; 0) = 0 for a.e. x ∈ 1 .
Combining the latter and the estimate with q ∈ L 1 ( 1 ), by Fatou's lemma, we get We use the compactness of the embedding V ⊂ H and (55) to get u n → u in H . Note that the functional ϕ is weakly lsc on V , being convex and lsc on V , which entails lim sup (ϕ(u) − ϕ(u n )) ≤ 0. The latter together with (58) allows to deduce from (57) that lim sup (A + B)u n , u n − u ≤ 0. Recalling that the operator A + B : V → V * is pseudomonotone, by a result in Sect. 2, we have Next, let v ∈ V be arbitrary. We will pass to the limit in the inequality (53). From (59), it follows Analogously to (58), we obtain lim sup and Exploiting (60), (61) and (62) from (53) in the limit as n → ∞, we obtain that u ∈ V solves Problem 3. Hence u ∈ S( f ) which completes the proof.
Next, we consider a distributed parameter control problem which can be realized through electromagnetic forces, see [19] and the references therein. We need the following hypotheses. Proof Let {( f n , u n )} be a minimizing sequence for problem (52). This means that f n ∈ U ad , u n ∈ S( f n ) and From H (U ad ) it is clear that the sequence { f n } stays in a bounded subset of V . We use the reflexivity of V and we may assume, by passing to a subsequence if necessary, that f n f * in U . The weak closedness of the set U ad entails f * ∈ U ad . By Theorem 8 we infer, by passing to a subsequence, that u u * in V , where u * ∈ S( f * ). By the weak lower semicontinuity of F, we have q ≤ F( f * , u * ) ≤ lim inf F( f n , u n ) = q. This completes the proof.
Applying [10,Theorem 4.6.7] and the continuity of the map V u → Du ∈ L 2 ( ; M d ), we deduce that if L : × R d × M d × R d → R is a measurable function such that Proof Under the notation (26), (27) and (30), problem P( f n , g n , k n ) for the sequence {u n } takes the form for all v ∈ V , where I : V → R is defined by For the sequence {u n } ⊂ V , we obtain an estimate analogous to (54) with M > 0 independent of n, since the sequence { f n } is uniformly norm bounded.
Then, we may assume that there exists u ∈ V such that, at least for a subsequence, we have and by the compactness of the trace operator, u n → u in L 2 ( 1 ; R d ) and u n (x) → u(x) a.e. x ∈ 1 , u n (x) R d ≤ ρ(x) a.e. x ∈ 1 (65) with ρ ∈ L 2 ( 1 ). It remains to show that u is a solution to problem P( f , g, k). The passage to the limit, as n → ∞, in (63) can be done as in the proof of Theorem 8. First, due the weak lower semicontinuity on V of the functional I , by (64), we have Since g n ≥ 0, I (u n ) ≥ 0, by (66) To this end, we choose v = u as a test function in problem (63) to get (A + B)u n , u − u n + 1 k n (u nτ ) j 0 (u nτ ; u τ − u nτ ) d By H (k)(iii), similarly as in (58) where q ∈ L 1 ( 1 ). We use (68), (71) and and then from (70) we deduce (69). Subsequently, let v ∈ V be arbitrary. We shall pass to the limit in (63). To this end, we exploit (67) Finally, we conclude that u ∈ V is a solution to problem P( f , g, k) which concludes the proof.
As a corollary from Theorem 10, we deduce that for g n → 0 and μ(r ) = μ 0 , k n = k, f n = f , the Bingham fluid tends to behave as a Newtonian one.

Final Comments
The variational formulation in Problem 3 does not involve the fluid pressure due to a property of the test functions. For a non zero yield limit, it is an interesting and challenging problem to recover the pressure from this variational formulation. An attempt to retrieve the pressure for the non-Newtonian fluid with Dirichlet boundary conditions can be found in [36]. Related results on this topic involve [34] and a more recent paper [31]. The problems for the time-dependent Bingham type models via the variational-hemivariational inequalities have not been considered in literature and are worth studying in the future.
It would be also of interest to extend the results of this paper to the case of a more general law S = G(D) with a given function G : M d → M d instead of D → μ( D )D in the Bingham constitutive relation (2). This will open a way to deal with various