Evolutionary Oseen Model for Generalized Newtonian Fluid with Multivalued Nonmonotone Friction Law

The paper deals with the non-stationary Oseen system of equations for the generalized Newtonian incompressible fluid with multivalued and nonmonotone frictional slip boundary conditions. First, we provide a result on existence of a unique solution to an abstract evolutionary inclusion involving the Clarke subdifferential term for a nonconvex function. We employ a method based on a surjectivity theorem for multivalued L-pseudomonotone operators. Then, we exploit the abstract result to prove the weak unique solvability of the Oseen system.


Introduction
In this paper we investigate the non-stationary Oseen system of equations which describes the flow of a viscous incompressible generalized Newtonian fluid and is governed by nonlinear multivalued and nonmonotone boundary conditions of frictional type. This type of problem occurs when in the nonstationary generalized Navier-Stokes equation the nonlinearity in the convective term is linearized by replacing the first argument by an already computed approximation. Such an approximation is used in implicit time discretization method applied together with a fixed point strategy, see e.g. [5,12] and the references therein.
We consider a nonlinear slip boundary condition which is described by the subdifferential of a nonconvex potential function. In order to deal with the nonconvex potential we exploit the notion of the generalized gradient of Clarke, see [2]. For this reason the weak formulation of the problem takes the form of a parabolic hemivariational inequality. If the potential generating the slip condition is a convex function, then the variational formulation of the problem is a variational inequality, see e.g. [5,9,10,14]. The stationary and non-stationary Oseen equations with homogeneous Dirichlet boundary condition were studied by the Galerkin method in [8] while stationary flow of non-Newtonian fluid with frictional boundary conditions have been recently treated in [23].
The mathematical theory of hemivariational inequalities has started with a pioneering work of Panagiotopoulos [25] and has been extensively developed in the last 30 years mainly because of various applications. We refer to monographs [21,24,26,27] to the wealth of problems which solutions have been possible using the theory of hemivariational inequalities. The hemivariational inequalities which appear in problems of solid mechanics can be found in [11,13,15,18,29] and in problems of fluid mechanics

Preliminaries
In this section we shortly recall basic definitions on single-valued and multivalued operators in Banach spaces and on the Clarke subdifferential which are used in the sequel. More details on these topics can be found in monographs [2][3][4]32].
Let (X, · X ) be a reflexive Banach space with its topological dual denoted by X * . The notation ·, · X * ×X stands for the duality pairing of X * and X, and the space X endowed with the weak topology is denoted by w-X. Given a set D ⊂ X, we define D X = sup{ d X | d ∈ D}.
Consider a multivalued operator A : X → 2 X * . It is called bounded if it maps bounded sets into bounded ones. It is called coercive if either its domain D(A) = {u ∈ X | Au = ∅} is bounded or D(A) is unbounded and We recall the notion of pseudomonotonicity of a multivalued operator.

Definition 1.
Let A : X → 2 X * be a multivalued operator and L : D(L) ⊂ X → X * be a linear and maximal monotone operator. The operator A is called pseudomonotone with respect to L (or Lpseudomonotone) if the following conditions hold (a) for all u ∈ X the set Au is a nonempty, bounded, closed, and convex subset of X * . (b) A is upper semicontinuous (u.s.c.) from each finite dimensional subspace of X to X * endowed with the weak topology. (c) if {u n } ⊂ D(L), u n → u weakly in X, Lu n → Lu weakly in X * , u * n ∈ Au n is such that u * n → u * weakly in X * and lim sup u * n , u n − u X * ×X ≤ 0, then u * ∈ Au and u * n , u n X * ×X → u * , u X * ×X . We recall the following surjectivity result. It is stated in [4,Theorem 1.3.73] under the hypothesis that X is strictly convex. However, by passing to an equivalent norm on X, we may always assume that X is a strictly convex Banach space. Theorem 2. Let X be a reflexive Banach space, let L : D(L) ⊂ X → X * be a linear and maximal monotone operator. If A : X → 2 X * is bounded, coercive, and L-pseudomonotone, then L + A is surjective.
Consider a single-valued operator A : X → X * . The operator A is said to be demicontinuous if for all w ∈ X, the functional u → Au, w X * ×X is continuous, i.e., A is continuous as a mapping from X to w * -X * . It is monotone, if for all u, v ∈ X, we have Au − Av, u − v X * ×X ≥ 0. The operator A : X → X * is said to be bounded if it maps bounded subsets of X into bounded subsets of X * . Now, we recall the definitions of the generalized directional derivative and the generalized gradient of Clarke for a locally Lipschitz function h : E → R defined on a Banach space E. The generalized directional derivative of h at x ∈ E in the direction v ∈ E, denoted by h 0 (x; v), is defined by The generalized gradient of h at x ∈ E, denoted by ∂h(x), is a subset in the dual space E * given by The basic properties of the generalized directional derivative and the generalized gradient as well as the relations between the generalized directional derivative and classical notions of differentiability can be found in [2,3,21,24].

Subdifferential Inclusion of First Order
In this section we study the first order evolutionary inclusion which contains the Clarke subdifferential operator. Our aim is to prove an existence and uniqueness result.
We study the inclusion within the framework of an evolution (Gelfand) triple of spaces V ⊂ H ⊂ V * , where V is a reflexive and separable Banach space, H is a separable Hilbert space, the embedding V ⊂ H is continuous, and V is dense in H. Given 0 < T < ∞, 2 ≤ p < ∞ and 1/p + 1/q = 1, we introduce the spaces V = L p (0, T ; V ) and W = {w ∈ V | w ∈ V * }, where the time derivative w = ∂w/∂t is understood in the sense of vector-valued distributions. It follows from reflexivity of V that both V and its dual space V * = L q (0, T ; V * ) are reflexive Banach spaces. It is known that the space W endowed with the graph norm w W = w V + w V * is a separable and reflexive Banach space. Let where ·, · V * ×V stands for the duality brackets of the pair (V * , V ).
Let A : (0, T ) × V → V * and J : (0, T ) × V → R. We assume that J is locally Lipschitz in its second argument and we denote by ∂J the Clarke generalized gradient of J with respect to its second argument. Given f : (0, T ) → V * and w 0 ∈ V , we consider the following evolutionary inclusion.
In the study of Problem 3 we introduce the following definition.

Definition 4.
By a solution of Problem 3 we mean a function w ∈ W for which there exists w * ∈ V * such that w * (t) ∈ ∂J(t, w(t)) for a.e. t ∈ (0, T ), w (t) + A(t, w(t)) + w * (t) = f (t) for a.e. t ∈ (0, T ) and We need the following hypotheses on the data.
Step 1. Let A : V → V * and B : V → 2 V * be the Nemitsky operators corresponding to the translations of A(t, ·) and ∂J(t, ·) by the element w 0 , Define an operator L : The operator L is linear and maximal monotone (see [32,Proposition 32.10]). With these operators, we consider the following inclusion Then, w ∈ W is a solution of Problem 3 if and only if w − w 0 ∈ W satisfies (1).
In what follows we are going to apply Theorem 2 to prove that problem (1) has a solution. For this goal, we will show that F has the properties required in Theorem 2.
Combining these inequalities, we immediately deduce that F is a bounded operator, being the sum of two bounded operators.  (4), the Hölder inequality, and the inequality |a + b| p ≥ 2 1−p |a| p − |b| p for a, b ∈ R and 1 < p < ∞, we obtain (3) and (4), we obtain

Claim 2. F is a coercive operator. First, by H(A)(3) and (4), we have the following coercivity condition for A(t, ·)
In a consequence, we have Claim 3. F is a L-pseudomonotone operator. First, we show the following properties of the operator A.
For a proof of (5), let v n → v in V. By passing to a subsequence if necessary, by [21,Theorem 2 Thus Av n → Av weakly in V * . The standard argument shows that the entire sequence {Av n } converges weakly in V * to Av. This concludes the proof of property (5).+ The strong monotonicity property for A in (6) follows directly from hypothesis H(A) (4). We now prove that the operator F is pseudomonotone with respect to L, that is, it satisfies conditions (a)-(c) of Definition 1.
(a) For every v ∈ V, the set Fv is a nonempty, bounded, closed, and convex in V * . The fact that values of the operator F are nonempty and convex follows from the well known property (see [2,Proposition 2.1.2]) that values of ∂J(t, ·) are nonempty and convex subsets of V * for a.e. t ∈ (0, T ). From (2) and which proves the closedness of the set Bv. Hence, the set Fv is closed in V * for all v ∈ V, which concludes the proof of (a). ( where V * is endowed with the weak topology. In order to show this property, we apply [3,Proposition 4.1.4]. To this end, we prove that the weak inverse image Therefore with w * n ∈ Bv n and v * n ∈ D. Since {v n } is bounded in V and the operators A and B are bounded (cf. Claim 1), we know that {v * n } and {w * n } are both bounded in V * . Thus, at least for subsequences, we may suppose that By the definition of the operator B, we have w * n (t) ∈ ∂J(t, v n (t) + w 0 ) a.e. t ∈ (0, T ). (9) Taking into account the convergences (7) and w * n → w * weakly in V * , and the fact that ∂J(t, ·) is u.s.c. with closed and convex values, we can apply a convergence theorem found in [1, p. 60] to the inclusion (9) and deduce w * (t) ∈ ∂J(t, v(t) + w 0 ) a.e. t ∈ (0, T ). Hence, w * ∈ Bv.
By the demicontinuity of the operator A [cf. (5)], we have Av n → Av weakly in V * . Passing to the limit in (8), we obtain v * = Av + w * , where w * ∈ Bv and v * ∈ D. Therefore, v * ∈ Fv ∩ D, which implies v * ∈ F − (D). This proves that F − (D) is closed in V and concludes the proof of condition (b).
(c) The condition (c) of Definition 1 holds.
First, we observe that the operator F : V → 2 V * is strongly monotone. Indeed, by H(J)(4), we have for all w * i ∈ Bv i , v i ∈ V, i = 1, 2. This inequality together with (6) imply , it follows that the operator F is strongly monotone. Next, we prove that v n → v in V. From the strong monotonicity of F, we have Taking lim sup in the last inequality and using (10), we obtain Using the strong convergence of v n to v in V, and passing to a subsequence if necessary, we may Consequently, from v * n ∈ Fv n , we have v * n = Av n + w * n (13) with w * n ∈ Bv n , and thus w * n (t) ∈ ∂J(t, v n (t) + w 0 ) a.e. t ∈ (0, T ). By the boundedness of the operator B (cf. Claim 1), we can assume that w * n → w * weakly in V * with w * ∈ V * . Similarly as in the proof of condition (b), we use the convergences (12) and w * n → w * weakly in V * , and apply the convergence theorem of [1, p. 60] to obtain Hence, w * ∈ Bv. By the demicontinuity of the operator A [cf. (5)], we obtain Av n → Av weakly in V * . Passing to the limit in (13), we get v * = Av + w * . Since w * ∈ Bv, we have v * ∈ Fv. From v * n → v * weakly in V * and v n → v in V, we deduce (11), which concludes the proof of condition (c).
Having established Claims 1-3 and noting that the operator L is linear and maximal monotone, we are in a position to apply Theorem 2. We deduce that the problem (1) has at least one solution w ∈ D(L). Then, w + w 0 ∈ W is a solution of Problem 3. This concludes the proof of the existence part of the theorem.

. From hypotheses H(A)(4) and H(J)(4), we obtain
for all t ∈ [0, T ]. Hence, by the smallness condition in (H 2 ), it follows that w 1 = w 2 on [0, T ], i.e., a solution to Problem 3 is unique. JMFM Remark 6. It can be shown that the hypothesis H(J)(4) is equivalent to the following condition for all v 1 , v 2 ∈ V , a.e. t ∈ (0, T ). The latter has been recently used with p = 2 in the literature to prove the uniqueness of the solution to the variational-hemivariational inequality. We refer to [21,29] for examples of nonconvex functions which satisfy the condition (14). Furthermore, we note that when J(t, ·) is convex, then (14) holds with m J = 0, i.e., the condition (14) simplifies to the monotonicity of the (convex) subdifferential.

Physical Setting and Classical Formulation
In this section we introduce the physical setting of the fluid flow problem and provide the classical description of the Oseen model. The general physical setting is as follows. A viscous incompressible generalized Newtonian fluid occupies an open, bounded and connected set Ω in R d , d = 2, 3, with boundary Γ = ∂Ω supposed to be Lipschitz continuous. We denote by ν = (ν i ) the unit outward normal vector on Γ, by x = (x i ) ∈ Ω the position vector, and by t ∈ (0, T ) the time, where 0 < T < ∞. We also assume that the boundary Γ is composed of two sets Γ D and Γ C , with disjoint relatively open sets Γ D and Γ C such that |Γ D | > 0.
We deal with the following non-stationary Oseen model which is used for the flow of incompressible fluid. The non-stationary flow of an incompressible generalized Newtonian fluid may be described by the following conservation laws (cf. e.g. [17] for further details) Here u = u(x, t) and π = π(x, t) denote the velocity field and the pressure, respectively, and is the external (gravity) force field. The expression denotes the convective term, and the solenoidal (divergence free) condition div u = ∇ · u = 0 in Ω states that the motion of the fluid is incompressible. Here, b is a given convection field which has to be divergence-free. The symbols Div and div denote the divergence operators for tensor and vector valued functions S : Ω → S d and u : Ω × (0, T ) → R d defined by Div S = (S ij,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. From time to time, we suppress the explicit dependence of the quantities on the spatial variable x ∈ Ω ∪ Γ. The total stress tensor in the fluid is given by σ = −π I + S in Ω, where I denotes the identity matrix and S : Ω → S d is the extra (viscous) part of the stress tensor. The symmetric part of the velocity gradient D : Ω → S d is given by D(u) = 1 2 (∇u + ∇u ). We assume that the extra stress tensor S is related with the symmetric part of the velocity gradient D by means of a constitutive law S = S(x, D(u)) in Ω. Also, we mention that for S(x, D(u)) = D(u) the Eq. (15) reduces to the linear Oseen system. We complement the above system with boundary conditions. Our main interest lies in the contact and slip frictional boundary conditions on the surface Γ C . On the part Γ D of the boundary, the fluid adheres to the wall, and therefore, we consider, for simplicity, the homogeneous Dirichlet condition On the part Γ C , we decompose the velocity vector into the normal and tangential parts. We denote by u ν and u τ the normal and the tangential components of u on the boundary Γ C , i.e., u ν = u · ν and u τ = u−u ν ν. Similarly, for an extra stress tensor field S, we define its normal and tangential components Vol. 20 (2018) Evolutionary Oseen Model for Generalized Newtonian Fluid 1325 by S ν = (Sν) · ν and S τ = Sν − S ν ν, respectively. We assume that there is no flux condition through Γ C , so that the normal component of the velocity on this part of the boundary satisfies The tangential components of the stress tensor and the velocity are assumed to satisfy the following multivalued friction law −S τ ∈ ∂j(u τ ) on Γ C × (0, T ), where j : Γ C × R d → R is the prescibed function. Finally, the problem is suplemented by the initial condition u(x, 0) = u 0 in Ω. Under these notation, the classical formulation of the Oseen model for flow of incompressible fluid reads as follows. Problem P . Find a velocity field u : Ω × (0, T ) → R d , an extra stress tensor S : Ω → S d , and a pressure π : Ω × (0, T ) → R such that

Weak Formulation
In this section we present the variational formulation of Problem P . To this end, we introduce some additional notation and state the hypotheses on the data. We will treat the problem in the case d = 2 and d = 3. We use the symbol S d for the space of second order symmetric tensors on R d or, equivalently, the space of symmetric matrices of order d. The canonical inner products and the corresponding norms on R d and S d are given by respectively. Let 2 ≤ p < ∞ and consider the following spaces and for v ∈ V with c K > 0 (cf. [7,Theorem 4]), it follows that · W 1,p (Ω;R d ) and · V are the equivalent norms on V . Moreover, V is a reflexive separable Banach space, H is a separable Hilbert space, the embedding V ⊂ H is continuous and V is dense in H. This means that (V, H, V * ) is an evolution triple of spaces. Recall that in this setting, the space H is identified with its dual and we have V ⊂ H ⊂ V * with dense and continuous embeddings.
Next, analogously as in Sect. 3, we define the spaces V = L p (0, T ; V ), V * = L q (0, T, V * ) and W = {w ∈ V | w ∈ V * }. We also introduce the space Y = L p (Γ C ; R d ) and the trace operator γ : V → Y . Its norm is denoted by γ = γ L(V ;Y ) . For v ∈ V , we use the same symbol v for the trace of v on the boundary. Note that v ν = 0 and v τ = γv on Γ C for all v ∈ V .
Furthermore, we also recall that the following Green formulas hold (cf. [21, Theorems 2.24 and 2.25]) for smooth tensor S : Ω → S d and field v : Ω → R d , and for smooth vector fields w : Ω → R d and ψ : Ω → R.
In the study of Problem P , we will assume that the constitutive function S and the nonconvex potential j satisfy the following hypotheses.
is strongly monotone for a.e. x ∈ Ω, i.e., there exists m S > 0 such that ·) is relaxed monotone for a.e. x ∈ Γ C , i.e., there exist m j ≥ 0 such that (∂j(x, x ∈ Γ C . The exponent p and the external body force f satisfy the following assumptions.
Finally, the convection field and the initial condition satisfy the following hypothesis.
We now turn to the variational formulation of Problem P . In what follows, we assume that u, S and π are sufficiently smooth functions which solve (16)- (21). Let v ∈ V and t ∈ (0, T ). We multiply the Eq. (16) by v, integrate over Ω and use the Green formula (22) to find that

S(D(u(t)))ν · v dΓ
Vol. 20 (2018) Evolutionary Oseen Model for Generalized Newtonian Fluid 1327 Note that hypotheses H(f ) and H(p) guarantee that the integrals Ω ((b · ∇)u(t)) · v dx and Ω f (t) · v dx are well defined. Exploiting the Green formula (23) and conditions div v = 0 in Ω, v = 0 on Γ D , and v ν = 0 on Γ C , we obtain Next, from conditions v = 0 on Γ D and v ν = 0 on Γ C , it follows ∂Ω S (D(u(t) Hence and from (24), we deduce Using (20) and (21) we obtain the following variational formulation of Problem P .
We have the following existence and uniqueness result whose proof will be provided in the next section.
holds. Then Problem P V has a unique solution.

Proof of Theorem 7
We will apply an abstract result of Theorem 5. We introduce operators B, C : V → V * as follows for all u, v ∈ V . We will prove that the operator A = B + C : V → V * satisfies hypothesis H(A). Note that A is independent of t ∈ (0, T ). First, we establish some properties of the operator B. From H(S)(ii) and the Hölder inequality, we have Dudek JMFM for all u, v ∈ V . This implies that the operator B : V → V * is well defined and Bu V * ≤ a S |Ω| 1 q + u p−1 V for all u ∈ V , which implies the boundedness of B. Furthermore, condition H(S)(iv) implies for all u 1 , u 2 ∈ V , which means that the operator B is strongly monotone. Next, we show that the operator B : V → V * is continuous. To this end, let u n , u ∈ V and u n → u in V , as n → ∞. Hence D(u n ) → D(u) in L p (Ω; S d ). From [3, Proposition 2.2.41], by passing to a subsequence if necessary, we have D(u n )(x) → D(u)(x) in S d for a.e. x ∈ Ω, as n → ∞, and D(u n )(x) S d ≤ η(x) for a.e. x ∈ Ω with η ∈ L p (Ω). By assumption H(S)(i), we obtain S(x, D(u n )(x)) → S(x, D(u)(x)) in S d for a.e. x ∈ Ω. Hypothesis H(S)(ii) and the elementary inequality |x + y| r ≤ 2 r−1 (|x| r + |y| r ) for x, y ∈ R and 1 ≤ r < ∞ imply for a.e. x ∈ Ω. Hence, by the Lebesgue dominated convergence theorem, we infer S(·, D(u n )(·)) − S(·, D(u)(·)) q L q (Ω;S d ) → 0, as n → +∞. By the Hölder inequality, we have for all v ∈ V . Hence, it follows that Bu n converges to Bu in V * , as n → ∞. This proves that the operator B is continuous.
Summing up, the operator B : V → V * is well defined, bounded, strongly monotone, and continuous. Now, we establish some properties of the linear operator C : V → V * . First, we observe that from the following continuous embeddings for p ≥ d, we have V ⊂ L r (Ω; R d ) for any r ∈ (1, ∞), This implies that operator C : V → V * is well defined and continuous, so it is bounded. Moreover, we note that Vol. 20 (2018) Evolutionary Oseen Model for Generalized Newtonian Fluid 1329 for all u ∈ V . Then, we use density of V in V , and we get (29) for all u ∈ V . As C is linear, it follows that for u 1 , u 2 ∈ V . We deduce that operator C : V → V * is bounded, monotone and continuous. From properties established for operators B and C, we deduce that operator A is demicontinuous, strongly monotone with m A = m S c p K and Hence A safisfies conditions H(A). Next, we introduce the functional J : V → R by We will verify that J satisfies hypothesis H(J). Note that J is independent of t. By hypotheses H(j)(i) and (ii), it is clear that for the functional J defined by (31), conditions H(J) (1) and (2) hold. From [21,Theorem 3.47] and the relation ∂J(v) ⊂ ΓC ∂j(v τ ) dx for all v ∈ V , we deduce that the hypothesis H(J)(3) is satisfied with c 0 (t) = c j0 |Γ C | 1 q and c 1 = c j1 γ p . Next, from H(j)(iv), [21,Theorem 3.47], and the relation |ξ τ | ≤ ξ for all ξ ∈ R d , we have for all v 1 , v 2 ∈ V with m J = m j γ p . Hence, the subdifferential of ∂J(t, ·) is relaxed monotone, which implies condition H(J) (4). Conditions (H 1 ) and (H 2 ) are consequences of hypotheses H(f ) and (H 0 ), and (25), respectively.
By definition of the Clarke subdifferential, the problem (32) is equivalent to: find u ∈ W such that ⎧ ⎪ ⎨ ⎪ ⎩ for all v ∈ V, a.e. t ∈ (0, T ), u(0) = u 0 . Exploiting this relation in (33), we deduce that u ∈ W is a solution to Problem P V .
Finally, we show that a solution to Problem P V is unique. Let u 1 , u 2 ∈ W be solutions to Problem P V , that is