Steady flow of generalized Newtonian fluid with multivalued rheology and nonmonotone friction law

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Steady flow of generalized Newtonian fluid with multivalued rheology and nonmonotone friction law✩ Sylwia Dudek a, *, Piotr Kalita b , Stanisław Migórski c a

Institute of Mathematics, Faculty of Physics, Mathematics and Computer Science, Krakow University of Technology, ul. Warszawska 24, 31155 Krakow, Poland b Faculty of Mathematics and Computer Science, Jagiellonian University in Krakow, ul. Łojasiewicza 6, 30348 Krakow, Poland c Institute of Mathematics, Lodz University of Technology, ul. Wolczanska 215, 90924 Lodz, Poland

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Article history: Received 17 October 2016 Received in revised form 26 February 2017 Accepted 21 June 2017 Available online xxxx Keywords: Generalized Newtonian fluid Multivalued constitutive law Maximal monotone Clarke generalized gradient Frictional contact

a b s t r a c t We study the stationary incompressible flow of a generalized Newtonian fluid described by a nonlinear multivalued maximal monotone constitutive law and a multivalued nonmonotone frictional boundary condition. We provide results on the existence and uniqueness of a solution to the variational form of the problem. When the multivalued laws are of a subdifferential form, we prove the existence of a solution to a variational-hemivariational inequality for the flow’s velocity field. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction In this paper we study a mathematical model that describes the steady flow of incompressible generalized Newtonian fluids in a bounded domain. The fluid is assumed to satisfy a multivalued rheological law between the symmetric part of the velocity gradient and the extra stress tensor. On two components of the boundary of the domain, we assume different boundary conditions: the Dirichlet condition on the velocity on a part where the fluid is supposed to adhere to the boundary, and the multivalued friction condition which holds between the tangential components of the stress tensor and the velocity on the second part. The multivalued constitutive law considered in the model is given by an inhomogeneous (depending on the spatial variable) maximal monotone graph. Examples of such laws can be found in the modeling of various fluid behaviors in hemodynamics, glaciology, food rheology and polymer industry. In particular, our model includes the so-called Bingham and Herschel–Bulkley fluids, as well as the rigid perfectly plastic fluids (Section 6). The multivalued friction law is given by a closed multifunction with a suitable growth condition. The basic examples of such multifunctions are potential laws described by the Clarke subdifferential of a nonsmooth, nondifferentiable locally Lipschitz function. Mathematical problems concerning the flow of various kinds of fluids described by the Navier–Stokes equations and their generalizations have been studied in many papers, cf. e.g. [1–9]. Generalized Newtonian fluids which belong to the class of non-Newtonian fluids were treated in Málek et al. [7]. Multivalued power-law rheologies for incompressible flows were considered in [1,10]. The results on stationary non-Newtonian fluids governed by a nonlinear single-valued constitutive law ✩ Research supported in part by the National Science Centre of Poland under the Maestro Project no. DEC-2012/06/A/ST1/00262. Corresponding author. E-mail addresses: [email protected] (S. Dudek), [email protected] (P. Kalita), [email protected] (S. Migórski).

*

http://dx.doi.org/10.1016/j.camwa.2017.06.038 0898-1221/© 2017 Elsevier Ltd. All rights reserved.

Please cite this article in press as: S. Dudek, et al., Steady flow of generalized Newtonian fluid with multivalued rheology and nonmonotone friction law, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.06.038.

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and a multivalued nonmonotone subdifferential frictional boundary condition can be found in [11,12]. The frictional contact boundary conditions for fluid have been studied, for both convex and nonconvex potentials, for instance, by Duvaut and Lions [3], Fujita [4], Consiglieri [2], Migórski [13], Migórski and Ochal [14] for the steady flows, and by Fang and Han [15], Kalita and Łukaszewicz [5], Łukaszewicz [6], and Migórski and Ochal [16] for the evolutionary problems. In this paper, we extend the earlier results of [1,11] in two directions: we establish the existence result for a more general class of admissible constitutive relations and for nonmonotone multivalued boundary condition. Similarly as in [1,17], we assume that the constitutive law is given by a maximal monotone graph. In contrast to [11], the constitutive relation is multivalued which in a particular case of the subdifferential of a convex potential leads to a variational term in the inequality problem for the velocity field. Also, we assume that the classical adherence of the fluid to the boundary enclosing its flow holds only on a part of the boundary and our interest is in a friction nonsmooth multivalued condition. Our approach incorporates also models whose behavior is characterized in terms of variational-hemivariational inequalities, see Section 6. More details and examples on hemivariational inequalities can be found in [13,14,18–20]. The main result of this paper is Theorem 11 that states the conditions for the existence of a weak solution to the stationary flow problem for a generalized Newtonian fluid. We note that our method of proof is different from those in the aforementioned papers; it exploits a surjectivity result for multivalued pseudomonotone operators. However, the existence result holds, under a smallness hypothesis, in the case p ≥ 23d with d = 2, 3 and the problem to relax this hypothesis is left +d open. Furthermore, in Theorem 12, we deliver sufficient conditions for the uniqueness of solution in the case d = 2 and for p ∈ [2, ∞). The uniqueness of solution for the case d = 3 and other values of p remains another interesting open problem. The paper is organized as follows. In Section 2 we recall preliminary material. The physical setting for the flow problem together with its classical and weak formulations is provided in Section 3. In Section 4 and 5, the results on existence and uniqueness of weak solution are proved, respectively. Some examples of the constitutive laws and frictional boundary conditions are given in Section 6. 2. Preliminaries In this section we recall the basic notation and preliminary results needed in the following sections. Details can be found in [21–23]. Let (X , ∥ · ∥X ) be a Banach space and let X ∗ denote its topological dual. The notation ⟨·, ·⟩X ∗ ×X stands for the duality pairing of X ∗ and X . The space X endowed with its norm and weak topology, is denoted by X and w -X , respectively. We put ∥S ∥X = sup{ ∥s∥X | s ∈ S } for any subset S of X . By L(E , F ) we denote the class of linear and bounded operators from a Banach space E to a Banach space F . We recall the following definitions for single-valued operators. An operator A : X → X ∗ is called bounded if it maps bounded sets of X into bounded sets of X ∗ . It is called monotone if ⟨Au − Av, u − v⟩X ∗ ×X ≥ 0 for all u, v ∈ X . An operator A is called hemicontinuous if the real valued function t → ⟨A(u + t v ), w⟩X ∗ ×X is continuous on [0, 1] for any u, v , w ∈ X . An operator A : X → X ∗ is said to be pseudomonotone, if it is bounded and if un → u weakly in X and lim sup⟨Aun , un − u⟩X ∗ ×X ≤ 0 implies ⟨Au, u − v⟩X ∗ ×X ≤ lim inf⟨Aun , un − v⟩X ∗ ×X for all v ∈ X . Recall that if X is reflexive Banach space, then every bounded, hemicontinuous and monotone operator is pseudomonotone (cf. Proposition 27.6(a) in [23]). ∗ The next definitions hold for multivalued operators. For a multivalued operator A : X → 2X , its domain, range and graph ∗ ∗ are defined by D(A) = {x ∈ X | Ax ̸ = ∅}, R(A) = ∪{Ax | x ∈ X } and Gr(A) = {(x, x ) ∈ X × X | x∗ ∈ Ax}, respectively. The inverse operator to A, denoted by A−1 : X ∗ → 2X , is defined by A−1 x∗ = {x ∈ X | x∗ ∈ Ax} for x∗ ∈ X ∗ . Multivalued operator ∗ A : X → 2X is called monotone, if for all (x, x∗ ), (y, y∗ ) ∈ Gr(A), we have ⟨x∗ − y∗ , x − y⟩X ∗ ×X ≥ 0. An operator A is called maximal monotone, if it is monotone and if (x, x∗ ) ∈ X × X ∗ is such that

⟨

x∗ − y∗ , x − y

⟩

X ∗ ×X

≥ 0 for all (y, y∗ ) ∈ Gr(A),

then (x, x∗ ) ∈ Gr(A). The latter is equivalent to saying that Gr(A) is not properly contained in the graph of any other monotone multivalued operator. The following result provides a useful criterion for maximal monotonicity (cf. Proposition 1.3.11 of [22]). ∗

Theorem 1. Let X be a Banach space and A : X → 2X be a multivalued operator. If the following conditions hold (i) A is monotone; (ii) for all v ∈ X , the set Av is nonempty, weakly-∗-closed and convex; (iii) for all u, v ∈ X , λ → A(λu + (1 − λ)v ) has a closed graph in [0, 1] × X ∗ , where X ∗ is endowed with the weak-∗ topology; then A is a maximal monotone operator. The following definitions of pseudomonotonicity and generalized pseudomonotonicity (cf. Definition 1.3.63 of [22]) will be useful in the next sections. ∗

Definition 2. Let X be a reflexive Banach space. A multivalued operator A : X → 2X is called pseudomonotone, if the following conditions hold Please cite this article in press as: S. Dudek, et al., Steady flow of generalized Newtonian fluid with multivalued rheology and nonmonotone friction law, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.06.038.

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(i) the set Av is nonempty, bounded, closed and convex for all v ∈ X ; (ii) A is usc from each finite dimensional subspace of X into X ∗ endowed with the weak topology; (iii) if vn ∈ X , vn → v weakly in X and vn∗ ∈ Avn is such that lim sup ⟨vn∗ , vn − v⟩X ∗ ×X ≤ 0, then to each y ∈ V , there exists v ∗ (y) ∈ Av such that ⟨v ∗ (y), v − y⟩X ∗ ×X ≤ lim inf ⟨vn∗ , vn − y⟩X ∗ ×X . ∗

Definition 3. Let X be a reflexive Banach space. An operator A : X → 2X is called generalized pseudomonotone, if for every sequence vn → v weakly in X , vn∗ → v ∗ weakly in X ∗ , vn∗ ∈ Avn and lim sup ⟨vn∗ , vn − v⟩X ∗ ×X ≤ 0, we have v ∗ ∈ Av and ⟨vn∗ , vn ⟩X ∗ ×X → ⟨v ∗ , v⟩X ∗ ×X . It is known that the sum of pseudomonotone operators is pseudomonotone (cf. Proposition 1.3.68 in [22]). The following result relates the notions of pseudomonotonicity and generalized pseudomonotonicity (cf. Proposition 1.3.65 and 1.3.66 of [22]). ∗

Proposition 4. Let X be a reflexive Banach space. If A : X → 2X is a pseudomonotone operator, then it is generalized ∗ pseudomonotone. If A : X → 2X is a bounded, generalized pseudomonotone operator such that for each u ∈ X , Au is a nonempty, closed and convex subset of X ∗ , then A is pseudomonotone. The following result provides a useful relation between pseudomonotonicity and maximal monotonicity (cf. Corollary 1.3.67 in [22]). ∗

Theorem 5. Let X be a reflexive Banach space. If A : X → 2X is a maximal monotone operator with D(A) = X , then A is pseudomonotone. ∗

Definition 6. A multivalued operator A : X → 2X is said to be coercive if inf{ ⟨u∗ , u⟩X ∗ ×X | u∗ ∈ Au }

∥ u∥ X

→ ∞,

as ∥u∥X → ∞.

We state the surjectivity result which will be useful in Section 4 (cf. Theorem 1.3.70 of [22]). ∗

Theorem 7. Let X be a reflexive Banach space and the multivalued operator A : X → 2X be pseudomonotone and coercive. Then A is surjective, i.e., R(A) = X ∗ . We recall (cf. Chapter 3.9 in [21]) some properties of the Sobolev space W 1,p (Ω ; Rd ) needed in the sequel. Theorem 8. Let Ω ⊂ Rd be a bounded domain (open connected set) with Lipschitz boundary ∂ Ω and 1 < p < ∞. Then the ∗ space W 1,p (Ω ; Rd ) is separable and reflexive, the embedding W 1,p (Ω ; Rd ) ⊂ Lp (Ω ; Rd ) is continuous with

∗

p =

⎧ dp ⎪ ⎪ ⎪ ⎨d − p

if p < d

an arbitrary large real ⎪ ⎪ ⎪ ⎩ +∞

and the embedding W 1,p (Ω ; Rd ) ⊂ L

if p = d if p > d, p∗ −ε

(Ω ; Rd ) is compact for any ε ∈ (0, p∗ − 1].

Finally, the following result is a version of the Korn inequality which was established in a more general setting in [24]. Theorem 9. Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary ∂ Ω , Γ ⊂ ∂ Ω be a nonempty, relatively open set, and p > 1. Define E = { u ∈ W 1,p (Ω ; Rd ) | u = 0 on Γ },

∇ u + ∇ u⊤ ). Then there exists a constant cK > 0 dependent on Ω , Γ , d, and p such that (∫ ) 1p ∥D(u)∥pSd dx ≥ cK ∥u∥W 1,p (Ω ;Rd ) for all u ∈ E .

and denote D(u) =

1 ( 2

Ω

Here and below, we denote by Sd the space of d × d symmetric matrices. The canonical inner products and the corresponding norms on Rd and Sd are given, respectively, by

ξ · η = ξ i ηi , σ : τ = σij τij ,

∥ξ ∥Rd = (ξ · ξ )1/2 for all ξ , η ∈ Rd , ∥τ ∥Sd = (τ : τ )1/2 for all σ , τ ∈ Sd .

Following standard convention, the summation over two repeated indices is also applied. An n-dimensional measure of a set O ⊂ Rn is denoted by |O|. Please cite this article in press as: S. Dudek, et al., Steady flow of generalized Newtonian fluid with multivalued rheology and nonmonotone friction law, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.06.038.

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3. Physical setting and weak formulation In this section we provide the physical setting of the flow problem, state the hypotheses on its data and derive its weak formulation. The general physical setting is as follows. An incompressible viscous generalized Newtonian fluid occupies an open, bounded and connected set Ω in Rd , d = 2, 3, with boundary Γ = ∂ Ω assumed to be Lipschitz continuous. We denote by ν = (νi ) the outward unit normal vector at Γ . We also assume that the boundary Γ is composed of two sets Γ D and Γ C , with disjoint relatively open sets ΓD and ΓC such that |ΓD | > 0. The stationary flow of an incompressible generalized Newtonian fluid may be described by the conservation law (cf. e.g. [8] for more details)

− Div S + (u · ∇ )u + ∇π = f

in Ω ,

(1)

div u = 0

in Ω .

(2)

d Here u : Ω → Rd and π : Ω → R denote the ( velocity )field and the pressure, respectively, and f : Ω → R is a density

of external forces. The expression (u · ∇ )u =

∂ ui j=1 uj ∂ xj

∑d

i=1,d

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. The symbols Div and div denote the divergence operators for tensor and vector valued functions S : Ω → Sd and u : Ω → Rd defined by Div S = (Sij,j )

and

div u = (ui,i ),

and the index that follows a comma represents the partial derivative with respect to the corresponding component of x. The total stress tensor in the fluid is expressed by σ = −π I + S in Ω , where I denotes the identity matrix and S : Ω → Sd is the extra (viscous) part of the stress tensor. The symmetric part of the velocity gradient D : Ω → Sd is given by D(u) = 21 (∇ u + ∇ u⊤ ). Due to the principle of objectivity, the extra stress tensor S is related to the velocity gradient only through its symmetric part D by means of a multivalued constitutive law of the form S(x) ∈ T (x, D(u(x))) in Ω , where the d multivalued constitutive function T : Ω × Sd → 2S satisfies growth, coercivity, and maximal monotonicity conditions stated in hypothesis H(T ) below. On the part ΓD of the boundary, we consider the homogeneous Dirichlet condition u=0

on ΓD .

On the part ΓC , we decompose the velocity vector into the normal and tangential parts. We denote by uν and uτ the normal and the tangential components of u on the boundary ΓC , i.e., uν = u · ν and uτ = u − uν ν . Similarly, for the deviatoric stress tensor field S, we define its normal and tangential components by Sν = (S ν ) · ν and Sτ = S ν − Sν ν , respectively. We assume that there is no flux through ΓC so that the normal component of the velocity on this part of the boundary satisfies uν = 0

on ΓC .

The tangential components of the stress tensor and the velocity are assumed to satisfy the following multivalued friction law

− Sτ ∈ F (uτ ) on ΓC . The classical formulation of the steady flow of the generalized Newtonian fluid described above reads as follows. Problem P. Find a velocity field u : Ω → Rd , an extra stress tensor S : Ω → Sd and a pressure π : Ω → R such that

− Div S + (u · ∇ )u + ∇π = f S ∈ T (D(u)) div u = 0 uν = 0 − Sτ ∈ F (uτ ) u=0

in Ω ,

(3)

in Ω ,

(4)

in Ω ,

(5)

on ΓC ,

(6)

on ΓC ,

(7)

on ΓD .

(8)

Here and in what follows, we skip sometimes the dependence of various functions on the variable x ∈ Ω ∪ Γ . In the study of Problem P, we use the following spaces.

˜ ¯ ; Rd ) | div v = 0 in Ω , v = 0 on ΓD , vν = 0 on ΓC }, V = { v ∈ C ∞ (Ω V = closure of ˜ V in W 1,p (Ω ; Rd ).

(9)

The space V is equipped with the norm ∥v∥ = ∥v∥W 1,p (Ω ;Rd ) . On V we introduce also the norm given by

∥v∥V = ∥D(v )∥Lp (Ω ;Sd ) for v ∈ V . Please cite this article in press as: S. Dudek, et al., Steady flow of generalized Newtonian fluid with multivalued rheology and nonmonotone friction law, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.06.038.

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From the Korn inequality cK ∥v∥W 1,p (Ω ;Rd ) ≤ ∥D(v )∥Lp (Ω ;Sd ) for v ∈ V with cK > 0 (cf. Theorem 9), it follows that ∥ · ∥W 1,p (Ω ;Rd ) and ∥ · ∥V are the equivalent norms on V . We also introduce the trace operator γ : V → Lp (ΓC ; Rd ) with the norm denoted by ∥γ ∥ = ∥γ ∥L(V ;Lp (ΓC ;Rd )) . Note that vν = 0 and vτ = γ v on ΓC for all v ∈ V . We also recall that the following Green formulas hold (cf. [19])

∫ Ω

∫

S : D(v ) dx +

Div S · v dx =

Ω

∫ ∂Ω

S ν · v dΓ

(10)

for smooth tensor S : Ω → Sd and field v : Ω → Rd , and

∫ Ω

w · ∇ψ dx +

∫

div w ψ dx =

Ω

∫ ∂Ω

wν ψ dΓ ,

(11)

for smooth vector fields w : Ω → Rd and ψ : Ω → R. We now list the assumptions on the data of Problem P. We assume that the constitutive multifunction T , the friction multifunction F , the exponent p, and the external body force density f satisfy the following conditions. For the exponent p, let q denote the conjugate exponent to p defined by 1/p + 1/q = 1. d

H(T ): T : Ω × Sd → 2S is such that (i) T is measurable with respect to L(Ω ) ⊗ B(Sd ) and B(Sd ), i.e., T −1 (C ) = {(x, D) ∈ Ω × Sd | T (x, D) ∩ C ̸ = ∅} ∈ L(Ω ) ⊗ B(Sd ) for every open set C ⊆ Sd , where L(Ω ) denotes the σ -field of Lebesgue measurable subsets of Ω and B(Sd ) represents the σ -field of Borel subsets of Sd ; (ii) T (x, ·) is a maximal monotone operator for a.e. x ∈ Ω ; p−1 (iii) ∥T (x, D)∥Sd ≤ a1 (x) + a2 ∥D∥Sd for all D ∈ Sd , a.e. x ∈ Ω with a1 ∈ Lq (Ω ), a2 > 0; p (iv) S : D ≥ a3 ∥D∥Sd + a4 (x) for all S ∈ T (x, D), D ∈ Sd , a.e. x ∈ Ω with a3 > 0, a4 ∈ L1 (Ω ). d

H(F ): F : ΓC × Rd → 2R is such that (i) (ii) (iii) (iv)

F (x, s) is nonempty, closed and convex for all s ∈ Rd and a.e. x ∈ ΓC ; F (·, s) is measurable for all s ∈ Rd ; Gr(F (x, ·)) is closed in Rd × Rd for a.e. x ∈ ΓC ; ∥F (x, s)∥Rd ≤ k1 (x) + k2 ∥s∥pR−d 1 for all s ∈ Rd and a.e. x ∈ ΓC with k1 ∈ Lq (ΓC ) and k2 ≥ 0.

H(p):

3d 2+d

≤ p < ∞. if p ∈ [

dp

3d

, d), then f ∈ L dp−d+p (Ω ; Rd );

2+d if p = d, then f ∈ Lr (Ω ; Rd ) for some r ∈ (1, ∞); if p > d, then f ∈ L1 (Ω ; Rd ).

H(f ) :

Remark 10. By hypothesis H(T )(ii), we have that T (x, D) is closed and convex for all D ∈ Sd and a.e. x ∈ Ω (cf. Proposition 1.3.10 of [22]). Moreover, T (x, D) ̸ = ∅ for all D ∈ Sd and a.e. x ∈ Ω . Indeed, by condition H(T )(iii), the maximal monotone operator T is locally bounded, and therefore the inverse T −1 (x, ·) is surjective for a.e. x ∈ Ω (cf. Theorem 1.3.39 in [22]) which implies the statement. We now turn to the weak formulation of Problem P. It is obtained by eliminating the pressure in the integral equation. We assume in what follows that u, π and S are sufficiently smooth functions which solve (3)–(8). Let v ∈ V . We use Eq. (3) and the Green formula (10) to find that

∫ Ω

S : D(v ) dx +

∫ Ω

((u · ∇ )u) · v dx +

∫ Ω

∇π · v dx =

∫ Ω

f · v dx +

∫ ∂Ω

S ν · v dΓ .

Note that conditions H(p) and H(f ) guarantee that the integrals Ω ((u · ∇ )u) · v dx and Ω f · v dx are well defined. From the Green formula (11) and the conditions divv = 0 in Ω , v = 0 on ΓD , vν = 0 on ΓC , we get

∫

∫ Ω

∫

∇π · v dx = −

Ω

π divv dx +

∫ ΓD

πvν dΓ +

∫ ΓC

∫

πvν dΓ = 0.

We take into account the conditions v = 0 on ΓD and vν = 0 on ΓC , to see that

∫ ∂Ω

S ν · v dΓ =

∫ ΓD

S ν · v dΓ +

∫ ΓC

(Sν vν + Sτ vτ ) dΓ =

∫ ΓC

Sτ · vτ dΓ .

Summing up, we obtain

∫ Ω

S : D(v ) dx +

∫ Ω

((u · ∇ )u) · v dx =

∫ Ω

f · v dx +

∫ ΓC

Sτ · vτ dΓ .

Please cite this article in press as: S. Dudek, et al., Steady flow of generalized Newtonian fluid with multivalued rheology and nonmonotone friction law, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.06.038.

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The weak formulation of Problem P is the following. Problem PV . Find a velocity field u ∈ V such that

∫ Ω

S : D(v ) dx +

∫ Ω

((u · ∇ )u) · v dx +

∫ ΓC

η · vτ dΓ =

∫ Ω

f · v dx

(12)

hold for all v ∈ V with S(x) ∈ T (x, D(u(x))) for a.e. x ∈ Ω and η(x) ∈ F (x, uτ (x)) for a.e. x ∈ ΓC . The results on the existence of solutions for d = 2, 3 and uniqueness of solution in the case d = 2 to Problem PV will be provided in the next sections. 4. Existence of weak solutions In this section we will establish the existence of solutions to Problem PV for d = 2, 3. Theorem 11. Assume that H(T ), H(F ), H(p) and H(f ) hold, and a3 > k2 ∥γ ∥p .

(13)

Then Problem PV has a solution u ∈ V . ∗

Proof. It is based on the surjectivity result of Theorem 7. We introduce the operators A : V → 2V , B : V × V → V ∗ , ∗ B[·] : V → V ∗ and C : V → 2V defined by Au =

{

ξ ∈ V ∗ | ⟨ξ , v⟩V ∗ ×V =

⟨B(u, v ), w⟩V ∗ ×V = Cu = γ ∗ G(γ u)

∫ Ω

d

Ω

S : D(v ) dx for all v ∈ V with S(x) ∈ T (x, D(u(x)))

((u · ∇ )v ) · w dx, B[v] = B(v, v )

for u, v, w ∈ V ,

for u ∈ V , Lq (ΓC ;Rd )

where G : L (ΓC ; R ) → 2 p

∫

a.e. x ∈ Ω

}

for u ∈ V ,

(14)

(15)

(16) is given by

G(u) = { ζ ∈ Lq (ΓC ; Rd ) | ζ (x) ∈ F (x, u(x))

a.e. on ΓC }

for u ∈ Lp (ΓC ; Rd )

(17)

and γ denotes the adjoint operator to the trace operator γ . We will prove that the operator A + B[·] + C is pseudomonotone and coercive. To this end, we establish the following properties of the operators. ∗

Claim 1. The multivalued operator A is pseudomonotone. First, we prove that the operator A is maximal monotone. The proof of maximal monotonicity of A is inspired by the argument of [25]. We show that the operator A satisfies the hypotheses of Theorem 1. (i) The operator A is monotone. This follows from the fact that T (x, ·) is monotone for a.e. x ∈ Ω , cf. the hypothesis H(T )(ii). (ii) For every u ∈ V , Au is a nonempty and convex subset of V ∗ . From Remark 10, it follows that for u ∈ V , the set T (x, D(u(x))) is nonempty, convex and closed in Sd and for a.e. x ∈ Ω . Hence, by H(T )(i) and Theorem III.6 of [26], there exists a measurable selection S(x) ∈ T (x, D(u(x))) for a.e. x ∈ Ω . Thus, the estimate in H(T )(iii) implies

∥S ∥Lq (Ω ;Sd ) ≤ ∥a1 ∥Lq (Ω ) + a2 ∥u∥pV−1 , which entails S ∈ Lq (Ω ; Sd ), and yields the nonemptiness of Au. The convexity of the set Au follows from the convexity of values of the multifunction T . (iii) For every u ∈ V , Au is a weakly closed subset of V ∗ and the multivalued map λ → A(λv + (1 − λ)w ) has a closed graph in [0, 1] × V ∗ , where V ∗ is endowed with the weak topology, for all v , w ∈ V . The fact that Au is a weakly closed subset of V ∗ for u ∈ V follows from the second part of the statement. We will prove that the graph is sequentially compact, when V ∗ is endowed with weak topology, whence the closedness will follow. To this end, assume that (λn , ξn ) belongs to the graph. We will show that for a subsequence, we have (λn , ξn ) → (λ, ξ ) in [0, 1] × V ∗ and (λ, ξ ) lies in the graph. Since [0, 1] is compact, for a subsequence, we have λn → λ ∈ [0, 1]. Let v , w ∈ V and un = λn v + (1 − λn )w. Hence ξn ∈ Aun for all n ∈ N. It is clear that un → u in V with u = λv + (1 − λ)w. From H(T )(iii), it follows that the sequence ξn is bounded in V ∗ and, for a subsequence, ξn → ξ weakly in V ∗ . We show that ξ ∈ Au. The growth condition H(T )(iii) guarantees the existence of a sequence of selections Sn ∈ Lq (Ω ; Sd ) such that Sn (x) ∈ Please cite this article in press as: S. Dudek, et al., Steady flow of generalized Newtonian fluid with multivalued rheology and nonmonotone friction law, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.06.038.

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T (x, D(un (x))) for a.e. x ∈ Ω such that (14) holds with un , ξn , and Sn . From the estimate ∥Sn ∥Lq (Ω ;Sd ) ≤ ∥a1 ∥Lq (Ω ) + a2 ∥un ∥V , by passing to the subsequence, if necessary, we may suppose that Sn → S weakly in Lq (Ω ; Sd ) with S ∈ Lq (Ω ; Sd ). Therefore, to show that ξ ∈ Au, it remains to verify that S(x) ∈ T (x, D(u(x))) for a.e. x ∈ Ω . If we show that the set M = { x ∈ Ω | ∃ ζ ∈ Sd , ∃ K ∈ T (x, ζ ), (S(x) − K ) : (D(u(x)) − ζ ) < 0 }

has Lebesgue measure zero, then the hypothesis H(T )(ii) of maximal monotonicity of T implies that S(x) ∈ T (x, D(u(x))) a.e. on Ω . To show that |M| = 0, we write M = {x ∈ Ω | Rx ̸ = ∅}, where Rx = { (ζ , K ) ∈ Sd × Sd | K ∈ T (x, ζ ), (S(x) − K ) : (D(u(x)) − ζ ) < 0 }. Exploiting H(T )(i), by Theorem 3.5 of [19], we have that R is graph measurable, i.e., Gr(R) ∈ L(Ω ) ⊗ B(Sd ) ⊗ B(Sd ). Thus, by the Aumann–von Neumann Theorem (cf. Theorem 4.3.7 of [21]), there exists a measurable selection (ζ , K ) of R defined on M. Therefore K (x) ∈ T (x, ζ (x)) and (S(x) − K (x)) : (D(u(x)) − ζ (x)) < 0

(18)

for all x ∈ M. On the other hand, the hypothesis H(T )(ii) implies that T is monotone and (Sn (x) − K (x)) : (D(un (x)) − ζ (x)) ≥ 0 a.e. on M

(19)

for every n ∈ N. If |M| > 0, then by the Lusin theorem (cf. Theorem 2.5.15 in [21]), there exists a measurable compact subset M′ of M with |M′ | > 0 such that (ζ (x), K (x)) is bounded on M′ . We integrate (19) on M′ and pass to the limit, as n → ∞ to obtain

∫ M′

(S(x) − K (x)) : (D(u(x)) − ζ (x)) dx ≥ 0,

which is a contradiction with (18). Therefore, we deduce that |M| = 0 and conclude that ξ ∈ Au. Using the conditions (i)–(iii), by Theorem 1, it follows that the operator A is maximal monotone. Next, since D(A) = V (cf. Remark 10), from Theorem 5, we deduce that A is a pseudomonotone operator. This completes the proof of the claim. Claim 2. The single-valued operator B[·] is pseudomonotone. It follows from the hypothesis H(p) that the operator B[·] is well defined. Indeed, using the following continuous embeddings for p ≥ d, we haveV ⊂ Lr (Ω ; Rd ) for any r ∈ (1, ∞), ) [ 2p dp 3d , d , we haveV ⊂ L d−p (Ω ; Rd ) ⊂ L p−1 (Ω ; Rd ), for p ∈ d+2 we deduce that the bilinear operator B(·, ·) satisfies the inequality

⟨B(u, v ), w⟩V ∗ ×V ≤ ∥u∥

2p

L p−1 (Ω ;Rd )

∥∇v∥Lp (Ω ;Sd ) ∥w∥

≤ K ∥u∥V ∥v∥V ∥w∥V

2p

L p−1 (Ω ;Rd )

for all u, w, v ∈ V

(20)

with a constant K > 0. Hence B[·] is well defined, it can be easily proved that it is continuous. Moreover, we have

⟨B(u, v ), v⟩V ∗ ×V =

∫ Ω

=−

((u · ∇ )v ) · v dx =

1 2

∫ divu Ω

d ∑

∫ ∑ d Ω i,j=1

vj2 dx +

j=1

1 2

ui

∂vj vj dx = ∂ xi

∫ ΓD

uν

d ∑

∫ ∑ d Ω i,j=1

vj2 dΓ +

j=1

1 2

ui

2 ∂ vj dx ∂ xi 2

∫ ΓC

uν

d ∑

vj2 dΓ = 0

(21)

j=1

for all u, v ∈ ˜ V . Then, exploiting the density of ˜ V in V , we obtain that (21) holds for all u, v ∈ V . Furthermore, a direct calculation shows that

⟨B(u, v ), w⟩V ∗ ×V = −⟨B(u, w), v⟩V ∗ ×V

for all u, v, w ∈ V .

To prove that B[·] is pseudomonotone, in view of Proposition 4, it is sufficient to prove that un → u weakly in V implies that B[un ] → B[u] weakly in V ∗ , or that

⟨B(un , v ), un ⟩V ∗ ×V → ⟨B(u, v ), u⟩V ∗ ×V

for allv ∈ V .

In view of density of ˜ V in V , it is sufficient to obtain the result for v ∈ ˜ V , but this assertion follows from the compactness of the embedding V ⊂ L2 (Ω ; Rd ). This completes the proof of the claim. Please cite this article in press as: S. Dudek, et al., Steady flow of generalized Newtonian fluid with multivalued rheology and nonmonotone friction law, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.06.038.

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Claim 3. The multivalued operator C is pseudomonotone. First, using the hypothesis H(F ) and the argument similar to the one of Lemma 8 of [12], we obtain the following properties of the operator G. (i) G has nonempty and convex values; (ii) Gr(G) is sequentially closed in Lp (ΓC ; Rd ) × (w -Lq (ΓC ; Rd )) topology; p−1 (iii) ∥G(u)∥Lq (ΓC ;Rd ) ≤ ∥k1 ∥Lq (ΓC ) + k2 ∥u∥Lp (Γ ;Rd ) for all u ∈ Lp (ΓC ; Rd ). C

Next, we will prove that the operator C is pseudomonotone. To this end, we show that Cu is a nonempty, closed and convex subset of V ∗ , C is a bounded and generalized pseudomonotone operator, and use Proposition 4. It follows from the hypothesis (i) that the set γ ∗ G(γ u) ⊂ V ∗ is nonempty and convex for all u ∈ V . This set is also closed in V ∗ . Indeed, let un ∈ V , u∗n ∈ γ ∗ G(γ un ), u∗n → u∗ in V ∗ . Then u∗n = γ ∗ wn∗ and wn∗ ∈ G(γ un ). From (iii), the set G(γ un ) ⊂ Lq (ΓC ; Rd ) is bounded, and so we may assume, by passing to a subsequence, if necessary, that wn∗ → w ∗ weakly in Lq (ΓC ; Rd ), where w ∗ ∈ Lq (ΓC ; Rd ). From (ii), one deduces that w ∗ ∈ G(γ u). Also, from u∗n = γ ∗ wn∗ , we obtain u∗ = γ ∗ w ∗ . Therefore u∗ ∈ γ ∗ G(γ u). Hence, for any u ∈ V , the set γ ∗ G(γ u) is closed in V ∗ . q d ∗ Subsequently, by (iii), the operator G : Lp (ΓC ; Rd ) → 2L (ΓC ;R ) is bounded. It is also clear that C : V → 2V is a bounded operator. Finally, we prove that C is a generalized pseudomonotone operator. Suppose vn → v weakly in V , vn∗ → v ∗ weakly in V ∗ , vn∗ ∈ C (vn ) and lim sup ⟨vn∗ , vn − v⟩V ∗ ×V ≤ 0. We will prove that v ∗ ∈ C (v ) and ⟨vn∗ , vn ⟩V ∗ ×V → ⟨v ∗ , v⟩V ∗ ×V . Since the operator γ is compact, we obtain γ vn → γ v in Lp (ΓC ; Rd ). We can write vn∗ = γ ∗ wn∗ with wn∗ ∈ Lq (ΓC ; Rd ) and wn∗ ∈ G(γ vn ). Next, from (iii) and boundedness of ∥γ vn ∥Lp (ΓC ;Rd ) , we may assume, by passing to a subsequence, if necessary, that wn∗ → w ∗ weakly in Lq (ΓC ; Rd ) with w ∗ ∈ Lq (ΓC ; Rd ). The condition (ii) implies that w ∗ ∈ G(γ v ). We have v ∗ = γ ∗ w ∗ , and hence v ∗ ∈ γ ∗ G(γ v ) = C (v ). Moreover, we obtain

⟨vn∗ , vn ⟩V ∗ ×V = ⟨γ ∗ wn∗ , vn ⟩V ∗ ×V = ⟨wn∗ , γ vn ⟩Lq (ΓC ;Rd )×Lp (ΓC ;Rd ) → ⟨w∗ , γ v⟩Lq (ΓC ;Rd )×Lp (ΓC ;Rd ) = ⟨γ ∗ w∗ , v⟩V ∗ ×V = ⟨v ∗ , v⟩V ∗ ×V which completes the proof of the generalized pseudomonotonicity of the operator C . From Proposition 4, we deduce that C is pseudomonotone. This completes the proof of the claim. Exploiting Claims 1–3, by Proposition 1.3.68 in [22], we deduce that the operator A + B[·] + C is pseudomonotone. Next, we show that it is also coercive. By H(T )(iv) and (21), we have

⟨Au + B[u], u⟩V ∗ ×V = ⟨Au, u⟩V ∗ ×V + ⟨B[u], u⟩V ∗ ×V ≥

∫

p

( Ω

)

p

a3 ∥D(u)∥Sd + a4 (x) dx ≥ a3 ∥u∥V − ∥a4 ∥L1 (Ω )

(22)

for all u ∈ V . On the other hand, let u ∈ V and u∗ ∈ C (u). Thus u∗ = γ ∗ w ∗ and w ∗ ∈ G(γ u). From (i)–(iii) and fact that γ is a linear and bounded operator, we obtain

|⟨u∗ , u⟩V ∗ ×V | = |⟨w∗ , γ u⟩Lq (ΓC ;Rd )×Lp (ΓC ;Rd ) | ≤ ∥γ ∥∥w ∗ ∥Lq (ΓC ;Rd ) ∥u∥V ≤ ∥γ ∥(∥k1 ∥Lq (ΓC ) + k2 ∥γ ∥p−1 ∥u∥pV−1 )∥u∥V = ∥k1 ∥Lq (ΓC ) ∥γ ∥∥u∥V + k2 ∥γ ∥p ∥u∥pV .

(23)

Combining (22) and (23), we have

⟨(A + B[u] + C )u, u⟩V ∗ ×V ≥ a3 ∥u∥pV − k2 ∥γ ∥p ∥u∥pV − ∥k1 ∥Lq (ΓC ) ∥γ ∥∥u∥V − ∥a4 ∥L1 (Ω ) ≥ (a3 − k2 ∥γ ∥p )∥u∥pV − ∥k1 ∥Lq (ΓC ) ∥γ ∥∥u∥V − ∥a4 ∥L1 (Ω ) = β (∥u∥V )∥u∥V for all u ∈ V , where β (t) = (a3 − k2 ∥γ ∥p )t p−1 − ∥k1 ∥Lq (ΓC ) ∥γ ∥ − which completes the proof of the coercivity condition. Moreover, we introduce ˜ f ∈ V ∗ by

⟨˜ f , v⟩V ∗ ×V =

∫ Ω

f · v dx

∥a4 ∥L1 (Ω ) t

. Therefore, by (13), β (t) → +∞, as t → +∞,

for v ∈ V .

(24)

It follows directly from H(f ) and Theorem 8 that ˜ f is well defined. We are now in a position to apply Theorem 7. We infer that there exists u ∈ V solution to the following problem Au + B[u] + γ ∗ (G(γ u)) ∋ ˜ f. This means that

⟨ξ , v⟩V ∗ ×V + ⟨B[u], v⟩V ∗ ×V + ⟨η, γ v⟩Lq (ΓC ;Rd )×Lp (ΓC ;Rd ) = ⟨˜ f , v⟩V ∗ ×V Please cite this article in press as: S. Dudek, et al., Steady flow of generalized Newtonian fluid with multivalued rheology and nonmonotone friction law, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.06.038.

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for all v ∈ V , where ξ ∈ Au and η ∈ G(γ u). Taking into account the definitions of operators A and G and the fact that γ v = vτ for v ∈ V , it is clear that u ∈ V satisfies

∫ Ω

S : D(v ) dx +

∫ Ω

((u · ∇ )u) · v dx +

∫ ΓC

η · vτ dΓ =

∫ Ω

f · v dx

for all v ∈ V , where S(x) ∈ T (x, D(u(x))) for a.e. x ∈ Ω and η(x) ∈ F (x, uτ (x)) for a.e. x ∈ ΓC . This concludes the proof of the theorem. 5. Uniqueness of weak solution In this section we provide sufficient conditions for the uniqueness of solution to Problem PV in the case d = 2 and for p ∈ [2, ∞). The uniqueness of solution for the case d = 3 and other values of p remains an interesting open problem. We introduce the notation

{ ∥f ∥ =

∥f ∥Lr (Ω ;R2 ) ∥f ∥L1 (Ω ;R2 )

if p = 2, for some r ∈ (1, ∞) if p ∈ (2, ∞).

In the case p = 2 we denote the exponent conjugate to r by r ′ . From embedding theorem, we have if p = 2 ∥v∥Lr ′ (Ω ;R2 ) if p ∈ (2, ∞) ∥v∥L∞ (Ω ;R2 )

} ≤ C1 ∥v∥V

for v ∈ V , where the embedding constant is denoted by C1 in both cases. We introduce the space Z = closure of ˜ V in H 1 (Ω ; R2 ). From the Korn inequality for p = 2 (cf. Theorem 9), the space Z can be endowed with the norm ∥v∥Z = ∥D(v )∥L2 (Ω ;S2 ) . This norm is equivalent to the H 1 norm, so in particular, we have ∥v∥H 1 (Ω ;R2 ) ≤ C3 ∥v∥Z for v ∈ Z with a constant C3 > 0. Since p ≥ 2, it follows that ∥v∥Z ≤ C2 ∥v∥V for v ∈ V with a constant C2 > 0. Moreover, using the continuity of the embedding Z ⊂ L4 (Ω ; R2 ), the following estimate holds ∥v∥L4 (Ω ;R2 ) ≤ C4 ∥v∥Z for v ∈ Z with C4 > 0. Since the trace operator γ0 : Z → L2 (ΓC ; R2 ) is linear and continuous, we have ∥γ0 v∥L2 (ΓC ;R2 ) ≤ ∥γ0 ∥∥v∥Z for v ∈ Z (and also for v ∈ V ). Theorem 12. Let p ∈ [2, ∞) and H(F ) holds with a constant k1 > 0. Assume H(T ) with a4 = 0, H(f ) and (13). If, in addition, (i) (S1 − S2 ) : (D1 − D2 ) ≥ m1 ∥D1 − D2 ∥2S2 for all Si ∈ T (x, Di ), Di ∈ S2 , i = 1, 2 and a.e. x ∈ Ω with m1 > 0; (ii) (ξ1 − ξ2 ) · (s1 − s2 ) ≥ −m2 ∥s1 − s2 ∥2R2 for all ξi ∈ F (x, si ), si ∈ R2 , i = 1, 2, a.e. x ∈ Ω with m2 ≥ 0; and

(

1

C1 ∥f ∥ + k1 |ΓC | q ∥γ ∥ a3 − k2 ∥γ ∥

) p−1 1 <

p

m1 cK2 − m2 ∥γ0 ∥2

(25)

K

are satisfied with K = C2 C3 C42 , then the solution to Problem PV is unique. Proof. First, we show an estimate on the solution. Let u ∈ V be a solution to Problem PV , i.e., there are ξ ∈ Au and η ∈ G(γ u) such that

⟨ξ , v⟩V ∗ ×V + ⟨B[u], v⟩V ∗ ×V + ⟨η, γ v⟩Lq (ΓC ;R2 )×Lp (ΓC ;R2 ) = ⟨˜ f , v⟩V ∗ ×V 1

p−1

for all v ∈ V . Using H(F )(iv), we obtain ∥η∥Lq (ΓC ,R2 ) ≤ k1 |ΓC | q + k2 ∥γ ∥p−1 ∥u∥V

|⟨η, γ u⟩Lq (ΓC ;R2 )×Lp (ΓC ;R2 ) | =

∫ ΓC

and

η · uτ dΓ ≤ ∥η∥Lq (ΓC ,R2 ) ∥γ ∥ ∥u∥V 1

≤ (k1 |ΓC | q + k2 ∥γ ∥p−1 ∥u∥pV−1 ) ∥γ ∥ ∥u∥V . From this inequality and the property ⟨B[u], u⟩V ∗ ×V = 0, we obtain 1

p

p

a3 ∥u∥V ≤ ⟨ξ , u⟩V ∗ ×V ≤ C1 ∥f ∥ ∥u∥V + k1 |ΓC | q ∥γ ∥∥u∥V + k2 ∥γ ∥p ∥u∥V . Hence p

1

(a3 − k2 ∥γ ∥p )∥u∥V ≤ (C1 ∥f ∥ + k1 |ΓC | q ∥γ ∥)∥u∥V and, by the smallness condition (13), we have 1

∥u∥V ≤ R p−1 ,

(26)

Please cite this article in press as: S. Dudek, et al., Steady flow of generalized Newtonian fluid with multivalued rheology and nonmonotone friction law, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.06.038.

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where 1

R=

C1 ∥f ∥ + k1 |ΓC | q ∥γ ∥ a3 − k2 ∥γ ∥p

.

Next, let u1 , u2 ∈ V be two solutions to Problem PV , that is

⟨ξi , v⟩V ∗ ×V + ⟨B[ui ], v⟩V ∗ ×V + ⟨ηi , γ v⟩Lq (ΓC ;R2 )×Lp (ΓC ;R2 ) = ⟨˜ f , v⟩V ∗ ×V with ξi ∈ Aui and ηi ∈ G(γ ui ) a.e. on ΓC for i = 1, 2. It follows that

⟨ξ1 − ξ2 , u1 − u2 ⟩V ∗ ×V + ⟨B[u1 ] − B[u2 ], u1 − u2 ⟩V ∗ ×V + ⟨η1 − η2 , u1τ − u2τ ⟩Lq (ΓC ;R2 )×Lp (ΓC ;R2 ) = 0.

(27)

From hypothesis (i), we obtain

⟨ξ1 − ξ2 , u1 − u2 ⟩V ∗ ×V ≥ m1

∫ Ω

∥D(u1 ) − D(u2 )∥2S2 dx ≥ m1 cK2 ∥u1 − u2 ∥2Z .

Using hypothesis (ii) and the boundedness of the trace operator γ0 , we deduce

∫ ΓC

(η1 − η2 ) · (u1τ − u2τ ) dΓ ≥ −m2

∫ ΓC

∥u1τ − u2τ ∥2R2 dΓ ≥ −m2 ∥γ0 ∥2 ∥u1 − u2 ∥2Z .

(28)

Exploiting (26), we get

⏐ ⏐∫ ⏐ ⏐ |⟨B[u1 ] − B[u2 ], u1 − u2 ⟩V ∗ ×V | = ⏐⏐ ((u1 − u2 )∇ )u2 · (u1 − u2 ) dx⏐⏐ Ω

1

≤ ∥u2 ∥H 1 (Ω ;R2 ) ∥u1 − u2 ∥2L4 (Ω ;R2 ) ≤ K R p−1 ∥u1 − u2 ∥2Z

(29)

with constant K = C2 C3 C42 > 0. Combining (27)–(29), we infer 1

m1 cK2 ∥u1 − u2 ∥2Z − m2 ∥γ0 ∥2 ∥u1 − u2 ∥2Z ≤ K R p−1 ∥u1 − u2 ∥2Z and 1

(m1 cK2 − m2 ∥γ0 ∥2 − K R p−1 ) ∥u1 − u2 ∥2Z ≤ 0. Finally, using condition (25) in the last inequality, it follows that u1 = u2 . The proof is complete. 6. Examples In this section we provide examples of constitutive multifunctions T and friction multifunctions F that satisfy the conditions in Problem P. Typical examples of such maps are of subdifferential type. For this reason, we hereafter recall the notions of convex and Clarke subdifferentials, cf. [19,21,27]. Definition 13. Let ϕ : X → R ∪{+∞} be a proper, convex and lower semicontinuous function on a Banach space X . The mapping ∗ ∂ϕ : X → 2X defined by

∂ϕ (x) = {x∗ ∈ X ∗ | ⟨x∗ , v − x⟩ ≤ ϕ (v ) − ϕ (x) for all v ∈ X } is called the convex subdifferential of ϕ . An element x∗ ∈ ∂ϕ (x) (if any) is called a subgradient of ϕ in x. Definition 14. Let h : X → R be a locally Lipschitz function on a Banach space X . The generalized (Clarke) directional derivative of h at x ∈ X in the direction v ∈ X , denoted by h0 (x; v ), is defined by h0 (x; v ) = lim sup

y→x, λ↓0

h(y + λv ) − h(y)

λ

.

The generalized (Clarke) subdifferential of h at x, denoted by ∂ h(x), is a subset of the dual space X ∗ given by

∂ h(x) = { ζ ∈ X ∗ | h0 (x; v ) ≥ ⟨ζ , v⟩ for all v ∈ X }. d

d

We introduce two operators T : Ω × Sd → 2S and F : ΓC × Rd → 2R defined by T (x, D) = ∂ψ (x, D)

for D ∈ Sd , a.e. x ∈ Ω

and F (x, ξ ) = ∂ j(x, ξ )

for ξ ∈ Rd , a.e. x ∈ ΓC ,

Please cite this article in press as: S. Dudek, et al., Steady flow of generalized Newtonian fluid with multivalued rheology and nonmonotone friction law, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.06.038.

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where ∂ψ and ∂ j denote the convex subdifferential of ψ and the Clarke subdifferential of j, respectively, and both subdifferentials are taken with respect to the second variable. We introduce the following hypotheses on potentials. H(ψ ): ψ : Ω × Sd → R is a function such that (i) (ii) (iii) (iv)

ψ (·, D) is measurable for all D ∈ Sd ; ψ (x, ·) is a convex function for a.e. x ∈ Ω ; p−1 there exist d1 ∈ Lq (Ω ), d2 > 0 such that ∥∂ψ (x, ξ )∥Sd ≤ d1 (x) + d2 ∥ξ ∥Sd for all ξ ∈ Sd , a.e. x ∈ Ω ; p there exist d3 > 0 and d4 ∈ L1 (Ω ) such that ζ : ξ ≥ d3 ∥ξ ∥Sd + d4 (x) for all ζ ∈ ∂ψ (x, ξ ), ξ ∈ Sd , a.e. x ∈ Ω .

H(j): j : ΓC × Rd → R is such that (i) j(·, ξ ) is measurable for all ξ ∈ Rd ; (ii) j(x, ·) is locally Lipschitz for a.e. x ∈ ΓC ; p−1 (iii) there exist l1 ∈ Lq (ΓC ), l2 ≥ 0 such that ∥∂ j(x, ξ )∥Rd ≤ l1 (x) + l2 ∥ξ ∥Rd for all ξ ∈ Rd , a.e. x ∈ ΓC . Consider the following problem which has the form of a variational-hemivariational inequality. Problem ˜ P. Find u ∈ V such that

∫

∫

(ψ (x, D(v )) − ψ (x, D(u))) dx + ((u · ∇ )u) · (v − u) dx ∫ ∫ Ω 0 + j (x, uτ ; vτ − uτ ) dΓ ≥ f · (v − u) dx for all v ∈ V .

Ω

ΓC

Ω

It follows from the definition of the convex and the Clarke subdifferentials that every solution of Problem PV with such choice of multifunctions, is also a solution of Problem ˜ P. From Corollary 2.3 of [28], Proposition 3.44 of [19] and Proposition 1.20 of [20], we deduce that under H(ψ ) the multifunction T satisfies H(T ) with a1 (x) = d1 (x), a2 = d2 , a3 = d3 , a4 (x) = d4 (x) for a.e. x ∈ Ω . Moreover, from Proposition 3.44 and Theorem 3.47 of [19], we deduce that under H(j), the multifunction F satisfies H(F ) with k1 (x) = l1 (x) for a.e. x ∈ ΓC and k2 = l2 . Therefore, from Theorem 11, we deduce the following. Corollary 15. Assume that hypotheses H(ψ ), H(j), H(p), H(f ) and the smallness condition (13) hold with a3 = d3 and k2 = l2 . Then Problem ˜ P has a solution u ∈ V . Next, we comment on concrete examples of the multivalued constitutive and frictional laws of the form (4) and (7). Examples 1 and 2 of multivalued constitutive laws come from [10,17], where their further discussion can be found. Example 1. We provide a concrete example of the multivalued constitutive function T of the form (4). Given a critical value of the shear rate d∗ > 0, we define

⎧ 2 ⎪ ⎨= ν1 (∥D∥Sd )D, T (D) = ν2 (∥D∥2Sd )D, ⎪ ⎩ ∈ [ν1 (d2∗ ), ν2 (d2∗ )]D,

if

∥D∥Sd < d∗

if

∥D∥Sd > d∗

if

∥D∥Sd = d∗ ,

(30)

for D ∈ S , where ν1 : [0, d∗ ] → R+ and ν2 : [d∗ , ∞) → R+ are continuous functions such that ν1 (d∗ ) ≤ ν2 (d∗ ), and [·, ·] denotes the line segment. d

The interpretation of the above law is the following. Once the shear rate reaches a certain critical value d∗ , this critical shear rate initiates series of reactions that, within a very short time interval, changes the viscosity of the material abruptly. The choice (30) has been inspired by the study due to Anand and Rajagopal [29] who developed a single continuum hemodynamical model that is capable of capturing flow changes in blood due to platelet activation. The viscosities νi , i = 1, 2 in (30) can describe various power-law fluid responses (cf. [10]) and are of form

νi (∥D∥2Sd ) = αi (κi + ∥D∥2Sd )

ri −2 2

(31)

where κi ≥ 0, αi > 0 and ri ∈ (1, ∞) for i = 1, 2. Note that the multifunction T given by (30) with velocities of the form (31) fulfills the hypotheses H(T )(i)–(iv) with p = r2 , cf. [10,17]. Example 2. Consider the following multifunction (cf. [1]) for the power-law fluids of the form B(0, τ ∗ ), ∗ D

{ T (D) =

τ

∥D∥

+˜ T (D),

if

D=0

if

D ̸ = 0,

(32)

where B(0, τ ∗ ) is the open ball in Sd centered in 0 with radius τ ∗ , τ ∗ > 0 is the so-called yield stress, and 2 ˜ T (D) = 2ν ∗ ∥D∥r − d D,

S

Please cite this article in press as: S. Dudek, et al., Steady flow of generalized Newtonian fluid with multivalued rheology and nonmonotone friction law, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.06.038.

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for D ∈ Sd with ν ∗ > 0 and r ∈ (1, ∞). The case r = 2 in (32) corresponds to the so-called Bingham model and the choice r ̸ = 2 corresponds to the Herschel–Bulkley model. If ν ∗ = 0, we have the model of the rigid perfectly plastic fluid considered in [3, p. 281]. We note that the multifunction T of the form (32) satisfies assumptions H(T )(i)–(iv) with p = r and the law ∗ (32) has the subdifferential form with the potential given by ψ (D) = τ ∗ ∥D∥Sd + 2νr ∥D∥rSd . In the following example we recall several single-valued constitutive laws which were considered earlier in the literature and satisfy hypothesis H(T ). Example 3. Let the constitutive function T : Ω × Sd → Sd be of the form T (x, D) = ν (∥D∥Sd )D for D ∈ Sd , a.e. x ∈ Ω .

(33)

The following particular cases of (33) were considered in [7,8]: p−2

T (1) (x, D) = κ1 ∥D∥Sd D, T (2) (x, D) = κ1 (1 + ∥D∥Sd )p−2 D, T (3) (x, D) = κ1 (1 + ∥D∥2Sd ) T

(3+i)

p−2 2

D,

(x, D) = κ0 D + T (x, D) (i)

for D ∈ S , a.e. x ∈ Ω , where i = 1, 2, 3, 1 < p < ∞, and κ0 , κ1 are suitable positive viscosity parameters. The fluids characterized by the constitutive law (33) are called generalized Newtonian fluids (even if they are non-Newtonian ones). We recall that if ν (r) = ν0 for r ≥ 0, ν0 > 0 is a given viscosity constant of the fluid, then (33) reduces to T (x, D) = ν0 D which represents a linear Stokes’ law, and (1) and (2) turn into the well known Navier–Stokes system. An incompressible fluids described by Stokes’ law are called Newtonian fluids. Fluids that cannot be characterized by the Stokes’ law are usually called non-Newtonian fluids, cf. [7–9] and the references therein. For a result which relates the properties of the function ν in (33) with the properties of T stated in hypothesis H(T ), we refer to Lemma 21 in [11]. We conclude this section with two concrete examples of the multivalued friction law of the form (7) generated by convex and nonconvex potentials, respectively. d

Example 4. Let u0 ∈ Rd be a given velocity of the moving part of boundary ΓC and κ > 0 be a friction coefficient. Consider the potential j(x, ξ ) = κ∥ξ − u0 ∥Rd which is convex in ξ , and set

⎧ ¯ , κ ), ⎨B(0 ξ − u0 F (x, ξ ) = ∂ j(x, ξ ) = , ⎩κ ∥ξ − u0 ∥Rd

if ξ = u0 , if ξ ̸ = u0

for all ξ ∈ Rd , a.e. x ∈ ΓC . The function j satisfies H(j) with l1 = κ and l2 = 0. Then the multivalued friction law (7) has the following form uτ = u0 H⇒ ∥Sτ ∥Rd ≤ κ,

{

uτ ̸ = u0 H⇒

− Sτ = κ

uτ − u0

∥uτ − u0 ∥Rd

(34)

.

This law is the well-known Tresca friction law on ΓC , cf. Section 6.3 of [19] for a detailed discussion. The interpretation of the above law is the following. In the case where the velocity of the fluid equals the velocity of the moving boundary, the tangential stress is below a certain threshold value. In turn, if the slip between the velocity of the fluid and the velocity of boundary occurs, then the friction force is directed opposite to the slip velocity and its magnitude is determined by the slip rate value according to the coefficient κ . It is also possible to consider the case when u0 depends on the variable x ∈ ΓC and to use a nonconstant function κ which depends on ∥uτ − u0 ∥Rd to highlight that the friction coefficient depends on the slip velocity. In the latter, the Tresca law is called the slip-rate dependent friction law. Example 5. Let j : ΓC × Rd → R be defined by j(x, ξ ) = (a − 1) e−∥ξ ∥Rd + a ∥ξ ∥Rd for all ξ ∈ Rd , a.e. x ∈ ΓC with a constant a ∈ [0, 1). Then, using the chain rule and the generalized gradient formula (see Clarke [27]), we have

⎧ ¯ , 1), ⎨B(0 ) F (x, ξ ) = ∂ j(x, ξ ) = ( −∥ξ ∥Rd +a ⎩ (1 − a) e

if ξ = 0,

ξ , ∥ξ ∥Rd

if ξ ̸ = 0

for all ξ ∈ Rd , a.e. x ∈ Γ . The function j(x, ·) is nonconvex and satisfies H(j) with l1 = 1 and l2 = 0. Then the multivalued friction law (7) takes the following form uτ = 0 H⇒ ∥Sτ ∥Rd ≤ 1,

{

uτ ̸ = 0 H⇒ − Sτ = (1 − a) e−∥uτ ∥Rd + a

(

)

uτ

∥ uτ ∥ R d

.

(35)

Please cite this article in press as: S. Dudek, et al., Steady flow of generalized Newtonian fluid with multivalued rheology and nonmonotone friction law, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.06.038.

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Please cite this article in press as: S. Dudek, et al., Steady flow of generalized Newtonian fluid with multivalued rheology and nonmonotone friction law, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.06.038.