Distributed control for shear-thinning

non-Newtonian fluids

telma guerra1

Abstract. We consider optimal control problems of systems governed by quasi-

linear, stationary, incompressible Navier-Stokes equations with shear-dependent

viscosity in a two-dimensional or three-dimensional domain. We study a general

class of viscosity functions with shear-thinning behaviour. Our aim is to prove

the existence of a solution for the class of control problems and derive the first

order optimality conditions.

Keywords. Optimal control; Non-Newtonian fluids; Shear-thinning; Optima-

lity conditions.

AMS Subject Classification. 49K20, 76D55, 76A05 l

1 Introduction

This paper deals with the mathematical formulation and analysis of an optimal

control problem of a viscous and incompressible fluid. The goal is to control the

system through a distributed mechanical force leading the velocity to a given

target field. Both control and state variables are constrained to satisfy a system

with shear dependent viscosity which decreases when the shear rate grows. We

deal with a quasi-linear generalization of the stationary Navier-Stokes system

described as follows−∇ · (τ(Dy)) + y · ∇y +∇π = u in Ω,

∇ · y = 0 in Ω,

y = 0 on ∂Ω,

(1)

where y denotes the velocity field, π denotes the pressure, τ is the Cauchy stress

tensor, which is a non linear function of Dy, the symmetric part of the velocity

1Escola Superior de Tecnologia do Barreiro, IPS. E-mail:telma.guerra@estbarreiro.ips.pt.Centro de Matematica e Aplicacoes, FCT-UNL.

1

2

gradient ∇y, i.e.

Dy = 12

(∇y + (∇y)T

).

The vector u is the given body force and Ω ⊂ IRn (n = 2 or n = 3).

Let us denote by yu a solution of (1) corresponding to the control u and intro-

duce the following optimal control problem

(Pα)

Minimize J(u)subject to u ∈ Uad.

(2)

where the funcional J : L2(Ω)→ IR is given by

J(u) = 12

∫Ω

|yu − yd|2 dx+ ν2

∫Ω

|u|2 dx. (3)

Here yd denotes a fixed element of L2(Ω), ν is a non-negative constant and Uad

is the set of admissible controls.

The class of fluids considered in this paper is described by partial differential

equations of the quasi-linear type that generalize the Navier-Stokes system re-

placing the Laplace operator by the divergence of τ . Such kind of fluids can be

frequently found in engineering literature (see e.g. [7], [20] and [17]). The first

mathematical analysis of such equations is due to the work of Ladyzhenskya [15]

and Lions [16] who proved the existence of weak solutions using compactness

arguments and the theory of monotone operators. We also suggest [18] for a

detailed discussion. Nowadays, we found a large range of literature containing

several results concerning the existence, uniqueness and regularity properties of

solutions. For this subject, in dimension n ≥ 2 we want to emphasize the work

of Crispo and Grisanti [10] which is related to our work. Under suitable small-

ness assumptions on the force term and without any restriction on the range of

α (1 < α < 2), the authors prove Holder continuity of the velocity field up to

the boundary. These results were extended by the same authors in [11] to the

case where p is a continuous and bounded function in Ω. Other papers had great

importance in the regularity up to the boundary in three-dimensional domains,

see [2, 19]. Beirao da Veiga in [2] obtained a great improvement of [19] by using

the classical Nirenberg translation method and opened the door for new ideas

in this research field. Those results were a posteriori extended to curvilinear

boundaries by the same author in [3]. New ideas were introduced in [4] and [5]

for the cases 1 < α < 2 and α ≥ 2 respectively, both without smallness assump-

tions on the force term. Other authors working on the subject that we want

3

to mention are: Kaplicky, Berselli, Diening, Roubıcek, Ruzicka, Frehse, Malek,

Posta , Necas, Steinhauser, Stara and Troltzsch [6, 12, 14, 19, 21, 22, 23, 24].

Control problems for non-Newtonian fluids were rarely studied in the past. For

the two-dimensional steady case we would like to mention the work of Casas

and Fernandez [8] and [9] and Slawig [25]. Both authors prove Gateaux diffe-

rentiability for quasi-linear elliptic equations, the first one considering the state

equation without the convective term and the divergence condition and the se-

cond one extending known results in the proof. Wachsmuth and Roubıcek [26]

studied the two-dimensional unsteady case and Gunzburger and Trenchea [13]

the three-dimensional coupled modified Navier-Stokes equations and Maxwell

equations. Our main work deals with the study of the control problem using

the new regularity results given in [10] in a three-dimensional domain, which is

the key point here. A explicit condition to conclude that the weak solution of

the problem (1) is in fact the strong solution proved in [10] is established and

Gateaux differentiability of the control-to-state mapping is obtained.

The paper is organized as follows. In Section 2 we fix de notation and state

the assumptions that are going to be useful in the sequel. In Section 3 we

summarize the existence and uniqueness of weak solution of the sate equation

and prove that is in fact a strong solution, see [10]. In Section 4, we define the

set of admissible controls, Uad, and we study the optimal control problem and

existence of solution of (Pα) in Uad. Lipschitz estimates are proved in Section

5 and in Section 6 we study the linearized equation showing the conditions for

the existence and uniqueness of solution. Following [9] and [25] we show in

Section 7 the Gateaux differentiability of the control-to-state mapping. Finally,

in Section 8 we prove the existence and uniqueness of solution to the adjoint

system and in Section 9 we prove the first order optimality conditions for (Pα).

2 Assumptions and notations

We assume that the tensor τ : IRn×nsym −→ IRn×nsym has a potential, i.e. there exists

a function Φ ∈ C2(IR+0 , IR

+0 ) with Φ(0) = 0 such that

τij(η) =∂Φ(|η|2)

∂ηij= 2Φ′(|η|2) ηij , τ(0) = 0

for all η ∈ IRn×nsym

(here IRn×nsym consists of all symetric (n × n)-matrices). As

usual in this kind of problems, we assume the following assumptions:

4

A1 - There exists a positive constant δ such that for all i, j, k, ` = 1, · · · , d∣∣∣∣∂τk`(η)

∂ηij

∣∣∣∣ ≤ δ (1 + |η|2)α−2

2 for all η ∈ IRn×nsym .

A2 - There exists a positive constant ν such that

τ ′(η) : ζ : ζ =∑ijk`

∂τk`(η)

∂ηijζk`ζij ≥ ν

(1 + |η|2

)α−22 |ζ|2

for all η, ζ ∈ IRn×nsym .

These assumptions are commonly used in the literature and cover a wide range

of applications in non-Newtonian fluids. Typical prototypes of extra tensors

used in applications are

τ(η) = (1 + |η|2)α−2

2 η or τ(η) = (1 + |η|)α−2η,

where we can consider two cases for the exponent α. The case 1 < α < 2

corresponds to shear-thinning behaviour fluids, whereas α > 2 corresponds to

shear-thickening.

Assumptions A1-A2 imply the following standard properties for τ (see [18]):

• Continuity

|τ(η)| ≤ k0δ(1 + |η|2)α−2

2 |η|, (4)

with k0 ≡ k0(α, n).

• Coercivity

τ(η) : η ≥ να−1 |η|(|η|

α−1 − 1) ≥ ν2α (|η|α − 1). (5)

• Monotonicity

(τ(η)− τ(ζ)) : (η − ζ) ≥ ν(1 + |η|2 + |ζ|2)α−2

2 |η − ζ|2. (6)

In the sequel, we use functions with values in IRn or in IRn×n. The space of

infinitely differentiable functions with compact support in Ω will be denoted by

D(Ω). Its dual (the space of distributions) is denoted by D′(Ω). The standard

Sobolev spaces are denoted by Wk,α(Ω) (k ∈ IN and 1 < α < ∞), and their

norms by ‖ · ‖k,p. We set W0,α(Ω) ≡ Lα(Ω) and ‖ · ‖α ≡ ‖ · ‖Lα . The dual

5

space of W1,α0 (Ω) is denoted by W−1,α′(Ω) and its norm by ‖ · ‖−1,α′ . In order

to eliminate the pressure in the weak formulation of the state equation, we will

also use the space of divergence free functions defined by

V =ψ ∈ D(Ω) | ∇ · ψ = 0

,

and denote by Vα the closure of V with respect to the norm ‖∇ · ‖α, i.e.

Vα =ψ ∈W1,α

0 (Ω) | ∇ · ψ = 0.

We define Cm,γ(Ω), the subspace of Cm(Ω) of all functions with all derivatives

that satisfy the following Holder condition

[y]Cm,γ (Ω) ≡∑|α|=m

supx1,x2∈Ω,x1 6=x2

|Dαy(x1)−Dαy(x2)||x1−x2|γ < +∞,

for m a nonnegative integer and 0 < γ < 1. The subspace Cm,γ(Ω) is a Banach

space with norm

‖y‖Cm,γ(Ω) ≡m∑|α|=0

‖Dαy‖∞ + [y]Cm,γ(Ω), (7)

Moreover, we recall two classical inequalities, Poincare and Korn:

‖y‖α ≤ CP,α1+CP,α

‖y‖1,α ≤ CP,α‖∇y‖α y ∈W1,α0 (Ω)

CK,α‖y‖1,α ≤ ‖Dy‖α y ∈W1,α0 (Ω).

We will denote by CP and CK the constants corresponding to α = 2 and considerCP,αCK,α

= CP,K,α and CPCK

= CP,K . We will also denote by CE the embedding

constant of H10(Ω) into L4(Ω), i.e.

‖y‖4 ≤ CE‖y‖1,2.

3 State equation

In this section we present a weak formulation of problem (1) and we summarize

the existence and uniqueness of weak solution of the state equation. The main

goal is to conclude that the weak solution is in fact a strong solution due to

regularity conditions made on y.

Definition 3.1. Let u ∈W−1,α′(Ω). A function y ∈ Vα is a weak solution of

(1) if

(τ(Dy), Dϕ) + (y · ∇y, ϕ) = 〈u, ϕ〉 for all ϕ ∈ Vα. (8)

6

First mathematical investigations of system (1) under conditions (4)-(6) were

performed by J. L. Lions [16] who proved the existence of a weak solution for

α ≥ 3nn+2 using compactness arguments and the theory of monotone operators.

The restriction on the exponent α ensures that the convective term belongs to

L1(Ω) and that we can choose test functions in the space Vα.

For the subsequent analysis, we recall some important results from the litera-

ture. For the convenience of the reader and in order to fix the notation, some

proofs will be given.

Lemma 3.2. Let y1 be in Vα and let y2, y3 be in W1,α0 (Ω), then

(y1 · ∇y2, y3) = −(y1 · ∇y3, y2) and (y1 · ∇y2, y2) = 0. (9)

Lemma 3.3. Consider 1 < α ≤ 2.

i) Let f ∈ L∞(Ω), g ∈ L1(Ω) and h ∈ L2(Ω) be non negative functions

satisfying

h2(x) ≤ f(x)g(x) a.e x ∈ Ω,

then

‖h‖22 ≤ ‖f‖∞‖g‖1;

ii) Let f ∈ Lα

2−α (Ω), g ∈ L1(Ω) and h ∈ Lα(Ω) be non negative functions

satisfying

h2(x) ≤ f(x)g(x) a.e x ∈ Ω,

then

‖h‖2α ≤ ‖f‖ α2−α‖g‖1.

Proof. The proof is left to the reader. 2

Proposition 3.4. Consider 1 < α ≤ 2.

i) Let y1 and y2 be in C1,γ(Ω), then

(τ(Dy1)− τ(Dy2), D(y1 − y2)) ≥ ν‖D(y1 − y2)‖221 + ‖Dy1‖2−α∞ + ‖Dy2‖2−α∞

(10)

ii) Let y1 and y2 be in Vα, then

(τ(Dy1)− τ(Dy2), D(y1 − y2)) ≥ ν‖D(y1 − y2)‖2α|Ω| 2−α2 + ‖Dy1‖2−αα + ‖Dy2‖2−αα

(11)

7

Proof. Using (6) and setting

f = (1 + |Dy1|2 + |Dy2|2)2−α

2

g =1

ν(τ(Dy1)− τ(Dy2)) : D(y1 − y2)

h = |D(y1 − y2)|,

due to Lemma 3.3 i) and 1 < α < 2 we have

‖D(y1 − y2)‖22

≤∥∥∥∥(1 + |Dy1|2 + |Dy2|2)

2−α2

∥∥∥∥∞

∥∥ 1ν (τ(Dy1)− τ(Dy2)) : D(y1 − y2)

∥∥1

≤ 1ν

∥∥(1 + |Dy1|2−α + |Dy2|2−α)∥∥∞ (τ(Dy1)− τ(Dy2), D(y1 − y2))

≤ 1ν (1 + ‖Dy1‖2−α∞ + ‖Dy2‖2−α∞ )(τ(Dy1)− τ(Dy2), D(y1 − y2))

which gives (10). Similarly using Lemma 3.3 ii), see [1] Proposition 2.8., we

obtain

‖D(y1 − y2)‖2α

≤∥∥∥∥(1 + |Dy1|2 + |Dy2|2)

2−α2

∥∥∥∥ α2−α

∥∥ 1ν (τ(Dy1)− τ(Dy2)) : D(y1 − y2)

∥∥1

≤ 1ν (|Ω|

α2 + ‖Dy1‖αα + ‖Dy2‖αα)

2−αα (τ(Dy1)− τ(Dy2), D(y1 − y2))

≤ 1ν (|Ω|

2−α2 + ‖Dy1‖α−2

α + ‖Dy2‖α−2α )(τ(Dy1)− τ(Dy2), D(y1 − y2))

which gives (11). 2

Proposition 3.5. Assume that A1-A2 are fulfilled with 3nn+2 ≤ α < 2. Then

for u ∈ L2(Ω) equation (1) admits at least a weak solution yu ∈ Vα and the

following estimate holds

‖Dyu‖α ≤ F(‖u‖2ν

), (12)

where F : IR+ → IR+ is defined by F (x) = (k1x)1

α−1 + k2x, where k1 and k2

are constants. Moreover, if u satisfies the following condition

F(‖u‖2ν

)(|Ω|

2−α2 + 2F

(‖u‖2ν

))< ν

k3, (13)

where k3 is a constant, the weak solution is unique.

Proof. For existence we recall [16]. For the reader’s convenience we present the

proof of uniqueness and estimate (12) (for details see [1], Theorems 3.1 and 3.2.)

To prove the estimate, let us set ϕ = yu in (8) and we use (9) to obtain

(τ(Dyu), Dyu) = (u, yu).

8

Taking into account that W1,α0 (Ω) → L2(Ω) for α ≥ 2n

n+2 and by using Poin-

care’s and Korn’s inequalities we have

|(u, yu)| ≤ ‖u‖−1,α′‖yu‖1,α ≤ 1CK,α‖u‖−1,α′‖Dyu‖α

≤ C2,α

CK,α‖u‖2‖Dyu‖α = k1‖u‖2‖Dyu‖α. (14)

where C2,α is the embedding constant and k1 =C2,α

CK,α. On the other hand, using

(11) with y1 = yu and y2 = 0 it follows

(τ(Dyu), Dyu) ≥ ν‖Dyu‖2α|Ω|

2−α2 + ‖Dyu‖2−αα

(15)

Combining (14) and (15) we have

‖Dyu‖α ≤ k1‖u‖2ν

(|Ω|

2−α2 + ‖Dyu‖2−αα

). (16)

Applying Young´s inequality we obtain

k1‖u‖2ν‖Dyu‖2−αα ≤ (α− 1)

(k1‖u‖2ν

) 1α−1

+ (2− α)‖Dyu‖α. (17)

and the estimate follows by combining (16) and (17). To prove the uniqueness

of a weak solution, we assume that y1 and y2 are two weak solutions to the

control u and set ϕ = y1 − y2 in the corresponding weak formulation to obtain

(τ(Dy1)− τ(Dy2), D(y1 − y2)) + (y1 · ∇y1 − y2 · ∇y2, y1 − y2) = 0 (18)

Due to (9), the convective term can be written as

(y1 · ∇y1 − y2 · ∇y2, y1 − y2)

= (y1 · ∇y1 + y2 · ∇y1 − y2 · ∇y1 − y2 · ∇y2, y1 − y2)

= ((y1 − y2) · ∇y1 + y2 · ∇(y1 − y2), y1 − y2)

= ((y1 − y2) · ∇y1, y1 − y2)

= −((y1 − y2) · ∇(y1 − y2), y1), (19)

and then we get

(τ(Dy1)− τ(Dy2), D(y1 − y2)) = ((y1 − y2) · ∇(y1 − y2), y1). (20)

Using (11) and estimate (12), we obtain

(τ(Dy1)− τ(Dy2), D(y1 − y2)) ≥ ν‖D(y1 − y2)‖2α|Ω| 2−α2 + ‖Dy1‖2−αα + ‖Dy2‖2−αα

≥ ν‖D(y1 − y2)‖2α|Ω| 2−α2 + 2F 2−α

(‖u‖2ν

) (21)

9

On the other hand, by using the Poincare and Korn inequalities and classical

embedding results we have

|((y1 − y2) · ∇(y1 − y2), y1)| = |((y1 − y2) · ∇y1, y1 − y2)|

≤ ‖y1 − y2‖22αα−1

‖∇y1‖α

≤ C2E,α‖(y1 − y2)‖21,α‖∇y1‖α

≤ k3‖D(y1 − y2)‖2α‖Dy1‖α

≤ k3‖D(y1 − y2)‖2αF(‖u‖2ν

). (22)

where CE,α is the embedding constant and k3 =C2E,α

C3K,α

. Combining (20), (21)

and (22) we deduce that(ν − k3F

(‖u‖2ν

)(|Ω|

2−α2 + 2F 2−α

(‖u‖2ν

)))‖D(y1 − y2)‖2α ≤ 0

and thus if

ν > k3F(‖u‖2ν

)(|Ω|

2−α2 + 2F 2−α

(‖u‖2ν

))then y1 ≡ y2. 2

Definition 3.6. Let u ∈ L2(Ω). A function y is a C1,γ-solution of problem (1)

if y ∈ C1,γ(Ω), for some γ ∈ (0, 1), ∇ · y = 0, y|∂Ω = 0 and it satisfies

(τ(Dy), Dϕ) + (y · ∇y, ϕ) = 〈u, ϕ〉 , for all ϕ ∈ V2(Ω). (23)

Next Proposition due to Crispo and Grisanti [10], gives an existence and uni-

queness result of a strong solution under suitable smallness assumptions on u,

but without assumptions on exponent α.

Proposition 3.7. Assume that 1 < α < 2 and q > n. Let Ω be a domain of

class C1,γ0 , γ0 = 1− nq and let be u ∈ Lq(Ω). If ‖u‖q ≤ ∆ where ∆ = ∆(γ0, n, α)

is a positive small constant in a suitable sense, then there exists a unique C1,γ

-solution, yu, of the problem (1) such that y ∈ C1,γ(Ω) for γ < γ0 and

‖yu‖C1,γ(Ω) ≤ k4‖u‖q. (24)

Remark 3.8. The authors prove that constants ∆ and k4 are given by

∆ ≡ min

1,1

c1(2r+4−αc(r + 4− α))1

r+3−α,

1

k4(1 + k4)2−α(2− α+ 2c)

,

k4 = 2c(r+4−αr+3−α

), (25)

10

where r = r(γ0, n, α) is a positive real number greater than 2, c1 and c are

suitable positive constants. The authors also prove that if we consider q > 2n,

Ω a domain of class C2 and ‖u‖q sufficiently small, the unique solution of the

problem belongs to W 2,2(Ω) ∩ C1,γ(Ω).

Let us consider y1 the C1,γ-solution of the problem (1) given by Proposition 3.7

and y2 ∈ Vα the weak solution given by Proposition 3.5. We prove in the next

Proposition that (13) is the explicit condition to have uniqueness.

Proposition 3.9. Assume that A1-A2 are fulfilled with 3nn+2 ≤ α < 2. Then

if conditions of Proposition 3.7 are verified and if y1 and y2 are C1,γ and weak

solutions of the problem (1) satisfying (13), we have y1 ≡ y2.

Proof. Obviously the C1,γ solution y1 is also a weak solution in the sense of

definition (3.1). Therefore, we can use test functions ϕ ∈ Vα and set ϕ = y1−y2

to obtain

(τ(Dy1)− τ(Dy2), D(y1 − y2)) + (y1 · ∇y1 − y2 · ∇y2, y1 − y2) = 0. (26)

Using (9) we can rewrite the convective term as

y1 · ∇y1 − y2 · ∇y2 = (y1 − y2) · ∇y1 + y2∇(y1 − y2)

and (26) is equivalent to

(τ(Dy1)− τ(Dy2), D(y1 − y2)) = −((y1 − y2) · ∇y1, y1 − y2).

Using (11) and estimate (12) once more we have,

(τ(Dy1)− τ(Dy2), D(y1 − y2)) ≥ ν‖D(y1 − y2)‖2α|Ω| 2−α2 + 2F 2−α

(‖u‖2ν

)On the other hand, by using the Holder, Poincare and Korn inequalities, Sobolev

injections and estimate (12) we have

|((y1 − y2) · ∇y1, y1 − y2)| ≤ k3F(‖u‖2ν

)‖D(y1 − y2)‖2α,

where k3 =C2E,α

C3K,α

is the embedding constant. Therefore,

ν‖D(y1 − y2)‖2α ≤(|Ω|

2−α2 + 2F 2−α

(‖u‖2ν

))k3F

2−α(‖u‖2ν

)‖D(y1 − y2)‖2α.

That is equivalent to(ν −

(|Ω|

2−α2 + 2F 2−α

(‖u‖2ν

))k3F

2−α(‖u‖2ν

))‖D(y1 − y2)‖2α ≤ 0

11

and thus if

ν >(|Ω|

2−α2 + 2F 2−α

(‖u‖2ν

))k3F

2−α(‖u‖2ν

)(27)

we have y1 ≡ y2. 2

4 Optimal control problem and existence of so-lution

In this section we establish the existence of solution of the optimal control

problem. The strong convergence of (yk)k to yu in W1,α0 (Ω) was proved in

[1], Proposition 4.1. The difference between those results and our case consists

only in the fact that we have better regularity results on y and together with

(24) it allow us to obtain an additional information on (yk)k, that is the strong

convergence to y in C(Ω). However this information is not necessary to be used.

For proving that (yk)k strongly converges to yu in W1,α0 (Ω) it is sufficient to

use convergence in Lebesgue spaces. For the reader’s convenience we present

the proof here.

We consider (Pα) introduced in (2) and define the set of admissible controls

Uad = v ∈ Lq(Ω) | ‖v‖q ≤ K

where

K = supx ∈ IR+|ν > k4C

2P,Kx(1 + k4x

2−α),

with k4 defined by (25).

Proposition 4.1. Assume that (uk)k>0 converges to u weakly in L2(Ω). Then

(yk)k converges strongly to yu in W 1,α0 (Ω).

Proof. The convergence of (uk)k>0 to u in the weak topology of L2(Ω) implies

that (uk)k>0 is uniformly bounded and

‖uk‖2 ≤M, for k > k0.

Due to Proposition 3.5, it follows that

‖Dyk‖α ≤ F(‖uk‖2ν

)≤ F (Mν ), for k > k0

and the sequence (yk)k is then bounded in Vα. The previous estimate, together

12

with (4) implies

‖τ(Dyk)‖α′

α′ ≤ (k0δ)α′∫

Ω

(1 + |Dyk|2)α−2

2 α′ |Dy|α′dx

≤ (k0δ)α′∫

Ω

(1 + |Dyk|2)α2 dx

≤ (k0δ)α′(|Ω|

α2 + ‖Dyk‖αα

)≤ (k0δ)

α′(|Ω|

α2 + Fα

(‖uk‖2ν

))≤ (k0δ)

α′(|Ω|

α2 + Fα

(Mν

))for k > k0

and the sequence (τ(Dyk))k is uniformly bounded in Lα′(Ω). Then there exists

a subsequence, still indexed by k, y ∈ Vα and τ ∈ Lα′(Ω) such that (yk)k>0

weakly converges to y in Vα and (τ(Dyk))k>0 weakly converges to τ ∈ Lα′(Ω).

Due to (24) and to the compact injection C1,γ(Ω) into C(Ω) we have that yk

strongly converges to y in C(Ω). Substituing in (8) we obtain

(τ(Dyk)− τ(Dy), Dϕ) + (yk · ∇yk − y · ∇y, ϕ) = (uk − u, ϕ), (28)

for all ϕ ∈ Vα. Since 3nn+2 < α < 2, by using Sobolev embeddings and regularity

results assumed on y, we have

|(yk · ∇yk − y · ∇y, ϕ)|

= |((yk − y) · ∇yk, ϕ) + (y · ∇(yk − y), ϕ)|

= |((yk − y) · ∇yk, ϕ)− (y · ∇ϕ, (yk − y))|

≤ |((yk − y) · ∇yk, ϕ)|+ |(y · ∇ϕ, (yk − y))|

≤(‖∇yk‖∞‖ϕ‖ 2α

α−1+ ‖y‖ 2α

α−1‖∇ϕ‖ 2α

α+1

)‖yk − y‖C(Ω)

→ 0 when k → +∞.

Hence, passing to the limit in

(τ(Dyk), Dϕ) + (yk · ∇yk, ϕ) = (uk, ϕ), for all ϕ ∈ Vα,

we obtain

(τ , Dϕ) + (y · ∇y, ϕ) = (u, ϕ) for all ϕ ∈ Vα, (29)

In particular, taking into account (9) we may write

(τ , Dy) = (τ , Dy) + (y · ∇y, y) = (u, y). (30)

On the other hand, the monotonocity assumption (6) gives

(τ(Dyk)− τ(Dϕ), D(yk)−Dϕ) ≥ 0 for all ϕ ∈ Vα. (31)

13

Since,

(τ(Dyk), Dyk) = (uk, yk),

substituing in (31), we obtain

(uk, yk)− (τ(Dyk), Dϕ)− (τ(Dϕ), Dyk −Dϕ) ≥ 0 for all ϕ ∈ Vα.

Passing to the limit it follows

(u, y)− (τ , Dϕ)− (τ(Dϕ), Dy −Dϕ) ≥ 0 for all ϕ ∈ Vα.

This inequality together with (30) implies that

(τ − τ(Dϕ), Dy −Dϕ) ≥ 0 for all ϕ ∈ Vα,

Aplying M-property [16] , we have

(τ , Dϕ) = (τ(Dy), Dϕ) for all ϕ ∈ Vα.1 (32)

Combining (29) and (32), we deduce that

(τ(Dy), Dϕ) + (y · ∇y, ϕ) = (u, ϕ) for all ϕ ∈ Vα.

Hence, y ≡ yu. Let us prove that (yk)k>0 strongly converges to yu ∈W1,α0 (Ω).

Substituing in the weak formulation of (1), setting ϕ = yk − yu and taking into

account (11) we obtain

(τ(Dyk)− τ(Dyu), D(yk − yu))

≥ ν

|Ω| 2−α2 + ‖Dyk‖2−αα + ‖Dyu‖2−αα

‖D(yk − yu)‖2α

≥ ν

|Ω| 2−α2 + F 2−α(‖uk‖2ν

)+ F 2−α

(‖u‖2ν

)‖D(yk − yu)‖2α

≥ ν

|Ω| 2−α2 + F 2−α(Mν

)+ F 2−α

(‖u‖2ν

)‖D(yk − yu)‖2α, for all k > k0.

Therefore, using (9) and classical embedding results, we obtain

ν

|Ω| 2−α2 + F 2−α(Mν

)+ F 2−α

(‖u‖2ν

) limk→+∞

‖D(yk − yu)‖2α

≤ limk→+∞

(τ(Dyk)− τ(Dyu), D(yk − yu))

= limk→+∞

((uk − u, yk − yu) + (yu · ∇yu − yk · ∇yk, yk − yu))

= limk→+∞

((uk − u, yk − yu)− ((yk − yu) · ∇yu, yk − yu))

= limk→+∞

((uk − u, yk − yu)− ‖yk − yu‖22α

α−1

‖∇yu‖α)

= 0.

which completes the proof. 2

14

Proposition 4.2. Assume that A1-A2 are fulfilled, with 1 < α ≤ 2. Then

(Pα) admits at least a solution.

Proof. Let (uk)k be a minimizing sequence and (yk)k the sequence of associated

states. Considering the properties of J , we obtain

λ2 ‖uk‖

22 ≤ J(uk)

implying that (uk)k is uniformly bounded in L2(Ω). From Proposition 4.1, we

deduce that (yk) converges strongly to yu. Taking into account the semiconti-

nuity of J , we deduce that

infkJ(·) ≤ J(u, yu) ≤ lim inf

kJ(uk, yk) ≤ inf

kJ(·)

that is u is a solution for (Pα). 2

5 Lipschitz estimates

Lemma 5.1. Let u1 and u2 be in Uad, and let yu1and yu2

be the corresponding

solutions of (1). Then the following estimate holds

‖D(yu1 − yu2)‖2 ≤ L(K)‖u1 − u2‖2, (33)

where L : IR+ → IR+ is a function defined by L(x) =CP,K(1 + 2k4x

2−α)

ν − k4xC2P,K(1 + 2k4x2−α)

.

Proof. Substituing in (8) we obtain

(τ(Dyu1)− τ(Dyu2), Dϕ) + (yu1 · ∇yu1 − yu2 · ∇yu2 , ϕ) = (u1 − u2, ϕ) (34)

for all ϕ ∈ Vα. Setting ϕ = yu1 − yu2 in (34), taking account (9) and the

definition of Uad and rewriting the convective term as

(yu1· ∇yu1

− yu2· ∇yu2

, yu1− yu2

)

= (yu1 · ∇yu1 + yu2 · ∇yu1 − yu2 · ∇yu1 − yu2 · ∇yu2 , yu1 − yu2)

= ((yu1− yu2

) · ∇yu1+ yu2

· ∇(yu1− yu2

), yu1− yu2

)

= ((yu1− yu2

) · ∇yu1, yu1

− yu2)

= −((yu1 − yu2) · ∇(yu1 − yu2), yu1), (35)

it follows

(τ(Dyu1)− τ(Dyu2), D(yu1 − yu2))

= (u1 − u2, yu1− yu2

) + ((yu1− yu2

) · ∇(yu1− yu2

), yu1).

15

Therefore

|(u1−u2, yu1−yu2

)| ≤ ‖u1−u2‖2‖yu1−yu2

‖2 ≤ CP,K‖u1−u2‖2‖D(yu1−yu2

)‖2,

and

|((yu1 − yu2) · ∇(yu1 − yu2), yu1)| = |((yu1 − yu2) · ∇yu1 , yu1 − yu2)|

≤ ‖(yu1− yu2

)2‖1‖∇yu1‖∞ ≤ ‖yu1

− yu2‖22‖∇yu1

‖∞

≤ C2P,Kk4‖D(yu1

− yu2)‖22‖u1‖q ≤ C2

P,Kk4K‖D(yu1− yu2

)‖22.

On the other hand, due to (10) and estimate (24)

(τ(Dyu1)− τ(Dyu2

), D(yu1− yu2

)) ≥ ν‖D(yu1 − yu2)‖221 + ‖Dyu1

‖2−α∞ + ‖Dyu2‖2−α∞

≥ ν‖D(yu1 − yu2)‖221 + k4‖u1‖2−αq + k4‖u2‖2−αq

≥ ν‖D(yu1− yu2

)‖221 + 2k4K2−α .

Therefore,

ν‖D(yu1− yu2

)‖221 + 2k4K2−α

≤ CP,K‖u1 − u2‖2‖D(yu1 − yu2)‖2 + C2P,Kk4K‖D(yu1 − yu2)‖22

which is equivalent to

‖D(yu1− yu2

)‖2 ≤CP,K(1 + 2k4K

2−α)

ν − C2P,Kk4K(1 + 2k4K2−α)

‖u1 − u2‖2

and the proof is complete. 2

6 Linearized equation

Before investigating the linearized equation we prove an auxiliary result.

Proposition 6.1. Let α be ]1, 2] and let z ∈ V2 and y ∈ C1,γ(Ω). Then∫Ω

(1 + |Dy|2)2−α

2 |Dz|2 dx ≥ ‖Dz‖221+‖Dy‖2−α∞

.

Proof. Since y and z belong to V2, by setting

f = (1 + |Dy|2)2−α

2 , g = |Dz|2(1 + |Dy|2)α−2

2 , h = |Dz|,

16

we can use Lemma 3.3 i) together with Holder’s inequality to obtain

‖Dz‖22 ≤∥∥∥∥(1 + |Dy|2)

2−α2

∥∥∥∥∞

∥∥∥∥|Dz|2(1 + |Dy|2)α−2

2

∥∥∥∥1

≤∥∥1 + |Dy|2−α

∥∥∞

∥∥∥∥|Dz|2(1 + |Dy|2)α−2

2

∥∥∥∥1

≤ (1 + ‖Dy‖2−α∞ )

∫Ω

(1 + |Dy|2)α−2

2 |Dz|2 dx

which gives the result. 2

We consider the linearized equation−∇ · (τ ′(Dyu) : Dz) + z · ∇yu + yu · ∇z +∇π = w in Ω,

∇ · z = 0 in Ω,

z = 0 on ∂Ω,

(36)

where u ∈ Uad, yu the corresponding solution of (1) and w ∈ L2(Ω).

Definition 6.2. Let w ∈ L2(Ω). A function z ∈ V2 is a weak solution of (36)

if

(τ ′(Dyu) : Dz,Dϕ) + (z · ∇yu + yu · ∇z, ϕ) = (w,ϕ), for all ϕ ∈ V2.

Proposition 6.3. Let u ∈ Uad and yu the corresponding solution of (1). For

w ∈ L2(Ω), problem (36) admits a unique solution zuw ∈ V2. Moreover the

following estimate holds

‖Dzuw‖2 ≤ L(‖u‖q)‖w‖2,

where L is defined in Lemma 5.1.

Proof. Consider the bilinear form defined by

B(z1, z2) = (τ ′(Dyu) : Dz1, Dz2) + (z1 · ∇yu + yu · ∇z1, z2).

Taking account (9) we have

B(z, z) = (τ ′(Dyu) : Dz,Dz) + (yu · ∇z, z) + (z · ∇yu, z)

= (τ ′(Dyu) : Dz,Dz) + (z · ∇yu, z),

for every z ∈ V2. Due to A1-A2, to Proposition 6.1 and (24) we deduce that

(τ ′(Dyu) : Dz,Dz) ≥ ν∫

Ω

(1 + |Dyu|2)α−2

2 |Dz|2 dx

≥ ν‖Dz‖221 + ‖Dyu‖2−α∞

≥ ν‖Dz‖221 + ‖∇yu‖2−α∞

≥ ν‖Dz‖221 + k4‖u‖2−αq

,

17

where k4 is given by (25). On the other hand, by using Holder inequality, (24),

Poincare’s and Korn’s inequalities we obtain

|(z · ∇yu, z| ≤ ‖z‖22‖∇yu‖∞ ≤ C2P,K‖Dz‖22‖∇yu‖∞

≤ k4C2P,K‖Dz‖22‖u‖q.

It follows that

B(z, z) ≥ ν‖Dz‖221 + k4‖u‖2−αq

− k4C2P,K‖Dz‖22‖u‖q

=

(ν

1 + k4‖u‖2−αq

− k4C2P,K‖u‖q

)‖Dz‖22 (37)

which shows that B(z, z) is coercive in V2. Let us now prove that B is conti-

nuous. With similar arguments, due to A1 and the fact that 3nn+2 < α < 2,

|τ ′(Dyu) : Dz1, Dz2)| ≤ δ∫

Ω

(1 + |Dyu|2)α−2

2 |Dz1||Dz2| dx

≤ δ∫

Ω

|Dz1||Dz2| dx

≤ δ‖Dz1‖2‖Dz2‖2.

On the other hand,

|(z1∇yu + yu∇z1, z2)|

≤ ‖z1‖2‖∇yu‖∞‖z2‖2 + ‖yu‖∞‖∇z1‖2‖z2‖2

≤ C2P ‖∇yu‖∞‖∇z1‖2‖∇z2‖2 + CP ‖yu‖∞‖∇z1‖2‖∇z2‖2

≤ C2P ‖∇yu‖∞‖z1‖1,2‖z2‖1,2 + CP ‖∇yu‖∞‖z1‖1,2‖z2‖1,2

≤ C2P,K‖∇yu‖∞‖Dz1‖2‖Dz2‖2 +

CP,KCK‖∇yu‖∞‖Dz1‖2‖Dz2‖2

≤ C2P,Kk4‖u‖q‖Dz1‖2‖Dz2‖2 +

CP,KCK

k4‖u‖q‖Dz1‖2‖Dz2‖2

= C2P,Kk4‖u‖q +

CP,KCK

k4‖u‖q)‖Dz1‖2‖Dz2‖2

=(C2P,K +

CP,KCK

)k4‖u‖q‖Dz1‖2‖Dz2‖2.

Therefore,

|B(z1, z2)| ≤(δ +

(C2P,K +

CP,KCK

)k4‖u‖q

)‖Dz1‖2‖Dz2‖2.

The bilinear form B is then continuous and coercive in V2. Applying the Lax-

Milgram Theorem, we deduce that problem (36) admits a unique solution zuw ∈

18

V2. Taking into account (37), we obtain(ν

1+k4‖u‖2−αq− k4C

2P,K‖u‖q

)‖Dzuw‖22 ≤ B(zuw, zuw) = (w, zuw)

≤ ‖w‖2‖zuw‖2

≤ CP,K‖w‖2‖Dzuw‖2,

which gives the result. 2

7 Differentiability of the state with respect tothe control

Following Casas and Fernandez [9], Slawig [25] and N. Arada [1], we show in

this section the Gateaux diferentiability of the control-to-state mapping. For

u and v in Uad and ρ ∈]0, 1[, set uρ = u + ρ(v − u), and let yρ ≡ yuρ be the

corresponding state and zρ =yρ−yuρ . Substituing in equation (8) we obtain

(τ(Dyρ)− τ(Dyu), Dϕ) + (yρ · ∇yρ − yu · ∇yu, ϕ) (38)

= (uρ − u, ϕ) = ρ(v − u, ϕ) for all ϕ ∈ V2.

Lemma 7.1. Let u and v be in Uad. Then the following estimate holds:

‖Dzρ‖2 ≤ L(K)‖v − u‖2, (39)

where L is given by Lemma 5.1.

Proof. It is a direct consequence of Lemma 5.1. 2

Lemma 7.2. Let zρk weakly converge to z in V2, for some sequence (ρk)k con-

verging to zero. Then

i)

limk→+∞

1

ρk(yρk · ∇yρk − yu · ∇yu, ϕ) = (z · ∇yu + yu · ∇z, ϕ),

ii)

limk→+∞

1ρk

(τ(Dyρk)− τ(Dyu), Dϕ) = (τ ′(Dyu) : Dz,Dϕ) ,

for all ϕ ∈ V.

19

Proof. Notice that∣∣∣∣ 1

ρk(yρk · ∇yρk − yu · ∇yu, ϕ)− (z · ∇yu + yu · ∇z, ϕ)

∣∣∣∣=

∣∣∣∣ 1

ρk((yρk − yu) · ∇yρk − yu · ∇(yρk − yu)), ϕ)− (z · ∇yu + yu · ∇z, ϕ)

∣∣∣∣= |(zρk · ∇yρk − yu · ∇zρk , ϕ)− (z · ∇yu + yu · ∇z, ϕ)|

= |(zρk · ∇yρk − yu · ∇zρk − z · ∇yu − yu · ∇z, ϕ)|

= |((zρk − z) · ∇yρk , ϕ)|+ |z · ∇(yρk − yu), ϕ)|

+ |(yu · ∇zρk , ϕ)− (yu · ∇z, ϕ)|

≤‖zρk − z‖4‖∇yρk‖∞‖ϕ‖4 + ‖z‖4‖∇(yρk − yu)‖2‖ϕ‖4

+ |(yu · ∇zρk , ϕ)− (yu · ∇z, ϕ)| .

The result is a consequence of the strong convergence of (yρk)k to yu in V2, the

weak convergence of (zρk)k to z in V2 and its strong convergence in L4(Ω) and

the fact that ∇yρk is continuous.

Let ϕ ∈ V be fixed. Using the mean value Theorem we obtain

1ρk

(τ(Dyρk)− τ(Dyu), Dϕ) = (τ ′ (σρk) : Dzρk , Dϕ) , (40)

where σρk(x) = s(x) (Dyρk(x)−Dyu(x)) + Dyu(x) with 0 < s(x) < 1. Con-

vergence of (σρk)k to Dyu in L2(Ω) and continuity of τ ′, imply that for every

ϕ ∈ V, we have

Dϕ : τ ′(σρk)→ Dϕ : τ ′(Dyu) a.e. in Ω.

On the other hand, due to A1, for all x ∈ Ω and i, j,m, l = 1, ..., n, we have

|(τ ′(σρk)(x))ijml| ≤ δ(1 + |σρk(x)|2

)α−22 ≤ δ

and thus

|(Dϕ(x) : τ ′(σρk)(x))ml| =

∣∣∣∣∣∣∑i,j

Dijϕ(x)(τ ′(σρk)(x))ijml

∣∣∣∣∣∣≤ δ

∑ij

|Dijϕ(x)| ≤ nδ|Dϕ|.

Due to the dominated convergence Theorem, we deduce that

limk→+∞

‖Dϕ : τ ′(σρk)−Dϕ : τ ′(Dyu)‖2 = 0.

This result together with the convergence of (Dzρk) to Dz in the weak topology

of L2(Ω) completes the proof.

2

20

Proposition 7.3. If (zρk)k weakly converges to z in V2 for some sequence (ρk)k

converging to zero, then z ≡ zuv − zuu, where zuw is the unique solution of the

linearized problem (36) corresponding to (yu, w). Moreover, (zρk)k converges

strongly to z in V2.

Proof. The first assertion is a direct consequence of Lemma 7.2, and of the

density of V in V2. Let us set

M = τ ′(Dy(x)), Mρ(x) = τ ′(σρ(x)),

where σρ(x) = s(x) (Dyρ(x)−Dyu(x)) +Dyu(x) with 0 < s(x) < 1 and

Ms(x) = M(x)+MT (x)2 , Ms

ρ(x) =Mρ(x)+MT

ρ (x)

2 .

Due to A2, the matrices Ms(x) and Msρ(x) are symmetric and positive definite.

Applying the Cholesky method to Ms(x) and Msρ(x), we deduce the existence

of lower triangular matrices L(x) and Lρ(x) such that

Ms(x) = L(x)LT (x) and Msρ(x) = Lρ(x)LTρ (x).

Using the mean value Theorem we obtain

1ρk

(τ(Dyρk)− τ(Dyu), Dϕ) = (τ ′ (σρk) : Dzρk , Dϕ) , (41)

where σρk(x) = s(x) (Dyρk(x)−Dyu(x)) + Dyu(x) with 0 < s(x) < 1. There-

fore, due to weak formulation (38) and to (41), we have∥∥LTρkDzρk∥∥2

2= (Mρk : Dzρk , Dzρk)

= − 1ρk

(yρk · ∇yρk − yu · ∇yu, zρk) + (v − u, zρk)

= −(zρk · ∇yρk + yu · ∇zρk , zρk) + (v − u, zρk)

= −(zρk · ∇yρk , zρk) + (v − u, zρk)

≤ ‖∇yρk‖∞‖z2ρk‖1 + ‖v − u‖2‖zρk‖2

≤ ‖∇yρk‖∞‖zρk‖22 + CP,K‖v − u‖2‖Dzρk‖2

≤ CP,K‖∇yρk‖∞‖Dzρk‖22 + CP,K‖v − u‖2‖Dzρk‖2

≤ CP,Kk4‖uρk‖qL2(K)‖v − u‖22 + CP,KL(K)‖v − u‖22

≤ CP,KL(K)‖v − u‖22 (k4‖uρk‖qL(K) + 1) . (42)

Last inequalities follows from (24) and (39) and hence the sequence (LTDzρk)k

is bounded in L2(Ω). On the other hand, due to A1 we have

|Lρk(x)|2 = |Mρk(x)| ≤ C(δ, n), for allx ∈ Ω.

21

Taking into account the convergence of (Dyρ)k to Dyu into L2(Ω), and the

continuity of τ ′, we deduce that Mρk(x) converges to M(x) and thus

Lρk(x) −→ L(x) for a.e. x ∈ Ω.

The dominated convergence Theorem then implies

Lρk −→ L strongly in L2(Ω) (43)

which together with the weak convergence of (zρk)k to z in V2, gives

LTρkDzρk −→ LTDz weakly in L2(Ω).

Moreover, taking into account (42), we deduce that∥∥LTDz∥∥2

2≤ lim inf

k

∥∥LTρkDzρk∥∥2

2≤ lim sup

k

∥∥LTρkDzρk∥∥2

2

= lim supk

(Mρk : Dzρk , Dzρk)

= lim supk

(− 1ρk

(yρk · ∇yρk − yu · ∇yu, zρk) + (v − u, zρk))

= lim supk

(−(zρk · ∇yρk + yu · ∇zρk , zρk) + (v − u, zρk))

= lim supk

(−(zρk · ∇yρk , zρk) + (v − u, zρk))

= −(z · ∇yu, z) + (v − u, z) = −(z · ∇yu − yu · ∇z, z) + (v − u, z)

= (M : Dz,Dz) =∥∥LTDz∥∥2

2.

Weak convergence together with norm convergence implies strong convergence

of (LTρkDzρk)k to LTDz in L2(Ω). Then there exists a subsequence, still indexed

by ρk, and a function H ∈ L2(Ω) such that∣∣LTρk(x)Dzρk(x)∣∣ ≤ H(x), for a.e. x ∈ Ω and k > k1,

LTρk(x)Dzρk(x) −→ LT (x)z(x), for a.e. x ∈ Ω.

Therefore, taking into account A2, we obtain

|Dzρk(x)|2 ≤Mρk(x) : Dzρk(x) : Dzρk(x)

= DzTρk(x)Mρk(x)Dzρk(x)

=∣∣LTρk(x)Dzρk(x)

∣∣2 ≤ H2(x), for a.e. x ∈ Ω and k > k1.

Since (43) implies(LTρk(x)

)−1 −→(LT (x)

)−1for a.e. x ∈ Ω,

it follows that

Dzρk(x) =(LTρk(x)

)−1LTρk(x)Dzρk(x) −→

(LT (x)

)−1LT (x)Dz(x) = Dz(x),

22

a.e. x in Ω. The conclusion follows by applying the dominated convergence

Theorem. 2

Proposition 7.4. Let G : Uad ⊂ L2(Ω) −→ V2 be the functional defined by

G(u) = yu, where yu is the solution of (1). Then G is Gateaux diferentiable

at u and its derivative in direction v − u is given by (G)′(u)(v − u) = z, where

z ∈ V2 is the unique solution of the problem (36) corresponding to (yu, v − u).

Moreover, we have

(J)′(u)(v − u) = (z, yu − yd) + ν(u, v − u). (44)

Proof. Let uρ = u + ρ(v − u), yρ ≡ yuρ and y ≡ yu for 0 < ρ < 1. Due to

Lemma 7.1, we deduce that(zρ =

yρ−yρ

)ρ

is bounded in V2. Then there exists

a subsequence (zρk)k and z ∈ V2 such that (zρk)k weakly converges to z in

V2. Due to Proposition 7.3, z is the solution of the linearized equation (36)

and (zρk)k strongly converges to z in V2. Taking into account Lemma 7.1, we

deduce that

‖Dz‖2 = limk‖Dzρk‖2 ≤ L(‖u‖q) ‖v − u‖2

which implies the continuity of (G)′(u) : L2(Ω) → V2, and therefore the

Gateaux differentiability of G. Finally, easy calculations show that

J ′(u)(v − u) = limρ→0

J(uρ)− J(u)

ρ

= limρ→0

(1

2

‖yρ − yd‖22 − ‖y − yd‖22ρ

+ν

2

‖uρ‖22 − ‖u‖22ρ

)= limρ→0

((zρ, y − yd) + ρ

2‖zρ‖22 + ν(u, v − u) + νρ

2 ‖v − u‖22

)= (z, y − yd) + ν(u, v − u)

which gives the result. 2

8 Adjoint equation

Let u ∈ Uad and let yu be the corresponding solution of (1). Consider the

adjoint system−∇ ·

(τ ′(Dyu)T : Dp

)+ (∇yu)T · p− yu · ∇p+∇π = f in Ω,

∇ · p = 0 in Ω,

p = 0 on ∂Ω,(45)

where f ∈ L2(Ω).

23

Definition 8.1. A funtion p is a weak solution of (45) if

(τ ′(Dyu) : Dϕ,Dp) + ((∇yu)T p− yu · ∇p, ϕ) = (f, ϕ)

for all ϕ ∈ V2.

Proposition 8.2. Let u ∈ Uad and let yu ∈ V2 be the corresponding solution

of (1). For f ∈ L2(Ω), system (45) admits a unique solution p in V2. Moreover,

the following estimate holds

‖Dp‖ ≤ L(‖u‖q)‖f‖2, (46)

where L is given by Lemma 5.1, and we have

(τ ′(Dyu) : Dϕ,Dp) + (ϕ · ∇yu + yu · ∇ϕ, p) = (f, ϕ), (47)

for all ϕ ∈ V2.

Proof. Existence and uniqueness of a solution as well as the estimate can be

obtained using arguments similar to those in the proof of Proposition 6.3. Mo-

reover, observing that

((∇yu)T p, ϕ) = (ϕ · ∇yu, p)

and

−(yu · ∇p, ϕ) = (yu · ∇ϕ, p),

we obtain (47). 2

9 First order optimality conditions for (Pα)

Let us now formulate the optimality conditions

Theorem 9.1. Assume that A1-A2 are fulfilled with 3nn+2 < α ≤ 2. Let u ∈

Uad be a solution of (Pα) and let y be the associated state. Then there then

exists a unique p ∈ V2, such that

(τ(Dy), Dϕ) + (y · ∇y) = (u, ϕ), for all ϕ ∈ Vα

(τ ′(Dy) : Dp,Dϕ)+((∇y)T p− y · ∇p, ϕ

)= (y−yd, ϕ), for all ϕ ∈ V2 (48)

(p+ νu,v − u) ≥ 0, for all v ∈ Uad, (49)

Moreover,

(τ ′(Dy) : Dp, p) + (p · ∇y, p) ≤ (y − yd, p).

24

Proof. Since J is Gateaux differenciable and Uad is convex, it is well known

that

J ′(u)(v − u) ≥ 0 for all v ∈ Uad. (50)

Due to Proposition 7.4 we have

J ′(u)(v − u) = (y − yd, zuv − zuu) + λ(u, v − u), (51)

where zuw is the solution of (36) corresponding to (y, w). Let p be the unique

solution of (45). Setting ϕ = zuv − zuu and taking into account the weak

formulation of the problem (36) and (47) we obtain

(y − yd, zuv − zuu) =(τ ′(Dy)T : Dp,D(zuv − zuu)

)+((∇y)T p− y · ∇p, zuv − zuu

)= (τ ′(Dy) : D(zuv − zuu), Dp)

+ ((zuv − zuu) · ∇y + y · ∇(zuv − zuu), p)

= (v − u, p). (52)

The result follows by combining (50), (51) and (52). 2

Acknowledgment. This work was partially supported by CMA/FCT/UNL,

under the project PEst-OE/MAT/UI0297/2011, and by PTDC/MAT109973/2009.

References

[1] N. Arada, Optimal control of shear-thinning flows, Technical Report, Cen-

tro de Matematica e Aplicacoes 2011, FCT-UNL, Portugal, Submitted.

[2] H. Beirao da Veiga, On the regularity of flows with Ladyzhenskaya shear-

dependent viscosity and slip or nonslip boundary conditions, Comm. Pure

App. Math 58 (4), (2005) pp. 552-577.

[3] H. Beirao da Veiga, On the Ladyzhenskaya-Smagorinsky turbulence mo-

del of the Navier-Stokes equations in smooth domains. The regularity pro-

blem., J. Eur. Math Soc. 11, (2009) pp. 127-167.

[4] H. Beirao da Veiga, On the global regularity of shear thinning flows in

smooth domains, J. Math. Anal. Appl. 339, (2009) pp. 335-360.

25

[5] H. Beirao da Veiga, P. Kaplicky, M. Ruzicka, Boundary Regularity

of Shear Thickening Flow, Math. Fluid Mech, 13 (3) (2011), Springer -

Base AG, pp. 387-404.

[6] L. C. Berselli, L. Diening, M. Ruzicka, Existence of Strong Solutions

for Incompressible Fluids with Shear Dependent Viscosities, J. Math. Fluid

Mech 12 (2010), pp. 101-132.

[7] R. B. Bird, R.C. Armstrong, O. Hassager, Dynamic of Polymer

Liquids, 2nd edition, Jonh Wiley, 1987.

[8] E. Casas, L.A. Fernandez, Boundary control of quasilinear elliptic equa-

tions, Rapport de Recherche 782, INRIA, 1988.

[9] E. Casas, L.A. Fernandez, Distrubuted control of systems governed by

a general class of quasilinear elliptic equations, J. Differential Equations 35

(1033), pp. 20–47.

[10] F. Crispo, C. R. Grisanti, On the Existence, Uniqueness and C1,γ(Ω)∩W 2,2(Ω) Regularity for a Class of Shear-Thinning Fluids, J. Math. Fluid

Mech. 10 (2008), pp. 445–487.

[11] F. Crispo, C. R. Grisanti, On the C1,γ(Ω) ∩W 2,2(Ω) regularity for a

class of electro-rheological fluids, J. Math. Anal. and App. 356 (2009), pp.

119-132.

[12] J. Frehse, J. Malek, M. Steinhauser, On analysis of steady flows

of fluids with shear-dependent viscosity based on the Lipschitz truncation

method, SIAM J. Math. Anal. 34 (5) (2003), pp. 1064–1083.

[13] M. Gunzburger, C. Trenchea, Analysis of an optimal control problem

for the three-dimensional coupled modified Navier-Stokes and Maxwell equa-

tions, J. Math. Anal. Appl. 333 (2007), pp. 295–310.

[14] P. Kaplicky, J. Malek, J. Stara, C1,α-solutions to a class of nonlinear

fluids in two dimensions-stationary Dirichlet problem, Zap. Nauchn. Sem.

POMI 259 (29) (1999), pp. 89–121.

[15] O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompres-

sible flow, Gordon and Beach, New York, 1969.

26

[16] J.L. Lions, Quelques methods de resolution des problemes aux limites non

lineaires, Dunod, Gauthier-Villars, Paris, 1969.

[17] J. Malek, K. L. Rajagopal, M. Ruzicka, Existence and regularity of

solutions and the stability of the rest state for fluids with shear dependent

viscosity, Math. Models Methods Appl. Sci. 5 (1995), pp. 789–812.

[18] J. Malek, J. Necas, J. Rokyta, M. Ruzicka, Weak and Measure-

valued Solutions to Evolutionary PDEs, Applied Mathematics and Mathe-

matical Computation, 13, Chapman and Hall, London, 1996.

[19] J. Malek, J. Necas, M. Ruzicka, On weak solutions to a class of non-

Newtonian incompressible fluids in bounded three-dimensional domains: the

case p ≥ 2, Adv. Differ. Equ. 6 (3), (2001), pp. 257-302.

[20] J. Malek, K. L. Rajagopal, Mathematical issues concerning the Navier-

Stokes equations and some of its generalizations, Handb. Differ. Equ., Evo-

lutionary equations, II (2005), Elsevier/ North-Holland, Amsterdam, pp.

371–459.

[21] J. Malek, T. Roubıcek, Optimization of steady flows for incompressible

viscous fluids, Nonlinear Applied Analysis (Eds. A. Sequiera, H. Beir ao da

Vega, J.H.Videman) Plenum Press, New York, (1999), pp.355-372.

[22] M. Posta, T. Roubıcek, Optimal control of Navier-Stokes equations by

Oseen approximation, (Preprint2007-013, Necas Centre, Prague) Compu-

ters and Mathematics with Applications, 53 (2007), 569-581.

[23] T. Roubıcek, Optimization of steady-state flow of incompressible fluids,

Proc. IFIP Conf. Analysis and Optimization of Differential Systems,

(V.Barbu, I.Lasiecka, D.Tiba, C.Varsan, eds.), Kluwer Academic Publ.,

Boston (2003), pp.357-368.

[24] T. Roubıcek, F.Troltzsch, Lipschitz stability of optimal controls for the

steady-state Navier-Stokes equations Technical Report, Institute of Mathe-

matics, TU Berlin, No.749-2002 Control and Cybernetics 32 (2003), pp.

683-705.

[25] T. Slawig, Distributed control for a class of non-Newtonian fluids, J. Dif-

ferential Equations, 219 (2005), pp. 116–143.

27

[26] D. Wachsmuth, T. Roubıcek, Optimal control of incompressible non-

Newtonian fluids, Z. Anal. Anwend 29 (2010), pp. 351–376.