The Boussinesq system with non-smooth boundary conditions : existence, relaxation and topology optimization.

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The Boussinesq system with non-smooth boundary conditions : existence, relaxation and topology

optimization.

Alexandre Vieira, Pierre-Henri Cocquet

To cite this version:

Alexandre Vieira, Pierre-Henri Cocquet. The Boussinesq system with non-smooth boundary condi-

tions : existence, relaxation and topology optimization.. 2021. �hal-03207923�

THE BOUSSINESQ SYSTEM WITH NON-SMOOTH BOUNDARY

1

CONDITIONS : EXISTENCE, RELAXATION AND TOPOLOGY

2

OPTIMIZATION.

3

ALEXANDRE VIEIRA^∗ AND PIERRE-HENRI COCQUET^∗†

4

Abstract. In this paper, we tackle a topology optimization problem which consists in finding 5

the optimal shape of a solid located inside a fluid that minimizes a given cost function. The motion 6

of the fluid is modeled thanks to the Boussinesq system which involves the unsteady Navier-Stokes 7

equation coupled to a heat equation. In order to cover several models presented in the literature, we 8

choose a non-smooth formulation for the outlet boundary conditions and an optimization parameter 9

of bounded variations. This paper aims at proving existence of solutions to the resulting equations, 10

along with the study of a relaxation scheme of the non-smooth conditions. A second part covers 11

the topology optimization problem itself for which we proved the existence of optimal solutions and 12

provides the definition of first order necessary optimality conditions.

13

Key words. Non-smooth boundary conditions, topology optimization, relaxation scheme, di- 14

rectional do-nothing boundary conditions 15

AMS subject classifications. 49K20, 49Q10, 76D03, 76D55 16

1. Introduction.

17

Directional do-nothing conditions. For many engineering applications, simula-

18

tions of flows coupled with the temperature are useful for predicting the behaviour

19

of physical designs before their manufacture, reducing the cost of the development

20

of new products. The relevance of the model and the adequacy with the experiment

21

therefore become important [16, 41, 46]. In this paper, we choose to model the flow

22

with the Boussinesq system which involves the Navier-Stokes equations coupled with

23

an energy equation. In most mathematical papers analyzing this model [8, 27, 47],

24

homogeneous Dirichlet boundary conditions are considered on the whole boundary.

25

This simplifies the mathematical analysis of the incompressible Navier-Stokes equa-

26

tion since the non-linear term vanishes after integrating by part hence simplifying the

27

derivation of a priori estimates [7, 21, 26, 47].

28

However, several applications use different boundary conditions that model inlet,

29

no-slip and outlet conditions [1]. Unlike the inlet and the no-slip conditions, the

30

outlet conditions are more subject to modelling choices. A popular choice consists in

31

using a do-nothing outlet condition (see e.g. [25, 34, 48]) which naturally comes from

32

integration by parts when defining a weak formulation of the Navier-Stokes equations.

33

However, since this outlet condition can not deal with re-entering flows, several papers

34

use a non-smooth outlet boundary conditions for their numerical simulations (see e.g.

35

[5, 23]). A focus on non-smooth outflow conditions when the temperature appears

36

can be found in [12, 23, 42, 43].

37

In particular, directional do-nothing (DDN) boundary conditions are non-smooth

38

boundary conditions that become popular. The idea is originally described in [13],

39

and several other mathematical studies followed [5, 9, 11]. These conditions were

40

considered especially for turbulent flows. In this situation, the flow may alternatively

41

∗Physique et Ing´enierie Math´ematique pour l’ ´Energie et l’Environnement (PIMENT), Univer- sit´e de la R´eunion, 2 rue Joseph Wetzell, 97490 Sainte-Clotilde, France. (Alexandre.Vieira@univ- reunion.fr)

†Laboratoire des Sciences de l’Ing´enieur Appliqu´ees `a la M´ecanique et au G´enie Electrique (SIAME), E2S-UPPA, Universit´e de Pau et des Pays de l’Adour, 64000 Pau, France (Pierre- Henri.Cocquet@univ-reunion.fr,Pierre-Henri.Cocquet@univ-pau.fr)

exit and re-enter the domain. These directional boundary conditions tries to capture

42

this phenomenon, while limiting the reflection. It is worth noting that other boundary

43

conditions can be used, namely the so-called local/global Bernouilli boundary condi-

44

tions [12, 23, 43]. The latter implies the do-nothing boundary condition is satisfied

45

for exiting fluid and that both the normal velocity gradient and the total pressure

46

vanish for re-entering fluid. Nevertheless, in this paper, we are going to used non-

47

smooth DDN boundary condition since they are easier to impose though a variational

48

formulation.

49

Concerning the mathematical study of Boussinesq system with directional do-

50

nothing conditions, the literature is rather scarce. To the best of our knowledge, we

51

only found [6, 15], where the steady case is studied in depth, but the unsteady case

52

only presents limited results. Indeed, while [15, p. 16, Theorem 3.2] gives existence

53

and uniqueness of a weak solution with additional regularity to the Boussinesq system

54

involving non-smooth boundary conditions at the inlet, it requires the source terms

55

and the physical constants (e.g. Reynolds, Grashof numbers) to be small enough. We

56

emphasize that these limitations comes from the proof which relies on a fixed-point

57

strategy. The first aim of this paper will then be to fill that gap by proving existence

58

and, in a two-dimensional setting, uniqueness of solutions for the unsteady Boussinesq

59

system with non-smooth DDN boundary condition on the outlet.

60

Topology optimization. On top of the previous considerations, this paper aims at

61

using these equations in a topology optimization (TO) framework. In fluid mechanics,

62

the term topology optimization refers to the problem of finding the shape of a solid

63

located inside a fluid that either minimizes or maximizes a given physical effect. There

64

exist various mathematical methods to deal with such problems that fall into the

65

class of PDE-constrained optimization, such as the topological asymptotic expansion

66

[3, 14, 40] or the shape optimization method [24, 38, 39]. In this paper, we choose to

67

locate the solid thanks to a penalization term added in the unsteady Navier-Stokes

68

equations, as exposed in [4]. However, the binary function introduced in [4] is usually

69

replaced by a smooth approximation, referred as interpolation function [43], in order

70

to be used in gradient-based optimization algorithms. We refer to the review papers

71

[1, 22] for many references that deal with numerical resolution of TO problems applied

72

to several different physical settings. However, as noted in [1, Section 4.7], most

73

problems tackling topology optimization for flows only focus on steady flows, and

74

time-dependant approaches are still rare. Furthermore, to the best of our knowledge,

75

no paper is dedicated to the mathematical study of unsteady TO problems involving

76

DDN boundary conditions, even though they are already used in numerical studies

77

[12, 23, 42, 43]. Therefore, a second goal of this paper will be to prove existence

78

of optimal solution to a TO problem involving Boussinesq system with non-smooth

79

DDN boundary conditions at the outlet.

80

First order optimality conditions. As hinted above, a gradient based method is

81

often used in order to compute an optimal solution of a TO problem. However, the

82

introduction of the non-smooth DDN boundary conditions implies that the control-

83

to-state mapping is no longer differentiable. The literature presents several ways to

84

deal with such PDE-constrained optimization problems. Most focus on elliptic equa-

85

tions, using subdifferential calculus [17, 30, 19] or as the limit of relaxation schemes

86

[18, 35, 45]. We may also cite [37] for a semilinear parabolic case and [49] which

87

involves the Maxwell equations. We emphasize that using directly a subdifferential

88

approach presents several drawbacks: the subdifferential of composite functions may

89

be hardly computed, and the result may be hardly enlightening nor used [17]. We will

90

therefore use a differentiable relaxation approach, as studied in [45]. First, we will

91

2

be able to use standard first order necessary optimality conditions since the relaxed

92

control-to-state mapping will be smooth. A convergence analysis will let us design

93

necessary optimality condition for the non-smooth problem. Secondly, we find this

94

approach more enlightening, as it may be used as a numerical scheme for solving the

95

TO problem.

96

1.1. Problem settings. Let Ω ⊂ R

^d

, d ∈ {2, 3} be a bounded open set with

97

Lipchitz boundary whose outward unitary normal is n. We assume the fluid occupies

98

a region Ω

f

⊂ Ω and that a solid is defined by a region Ω

s

such that Ω = Ω

f

∪ Ω

s

.

99

The penalized Boussinesq approximation (see e.g. [43] for the steady case) of the

100

Navier-Stokes equations coupled to convective heat transfer reads:

101

(1.1)

∇ · u = 0,

∂

t

θ + ∇ · (uθ) − ∇ · (Ck(α)∇θ) = 0, a.e. in Ω

∂

_t

u + (u · ∇)u − A∆u + ∇p − Bθe

_y

+ h(α)u = f, u(0) = u

0

(α), θ(0) = θ

0

(α),

102

where u denotes the velocity of the fluid, p the pressure and θ the temperature (all

103

dimensionless), u

0

(α), θ

0

(α) are initial conditions. In (1.1), A = Re

⁻¹

with Re being

104

the Reynolds number, B = Ri is the Richardson number and C = (Re Pr)

⁻¹

where Pr

105

is the Prandtl number. In a topology optimization problem, it is classical to introduce

106

a function α : x ∈ Ω 7→ α(x) ∈ R

⁺

as optimization parameter (see e.g. [1, 22]). The

107

function h(α) then penalizes the flow in order to mimic the presence of solid:

108

• if h ≡ 0, then one retrieves the classical Boussinesq approximation.

109

• if, for some s > s

0

and large enough α

max

, h : s ∈ [0, α

max

] 7→ h(s) ∈ [0, α

max

]

110

is a smooth function such that h(s) ≈ 0 for s ≤ s

0

and h(s) ≈ α

max

for s ≥ s

0

,

111

one retrieves the formulations used in topology optimization [1, 8, 43]. In the

112

sequel, we work in this setting since we wish to study a TO problem.

113

Since the classical Boussinesq problem is retrieved when h(α) = 0, the fluid zones

114

Ω

f

⊂ Ω and the solid ones Ω

s

⊂ Ω can be defined as Ω

s

:= {x ∈ Ω | α(x) < s

0

} , Ω

f

:=

115

{x ∈ Ω | α(x) > s

0

} , where α

max

> 0 is large enough to ensure the velocity u is small

116

enough for the Ω

s

above to be considered as a solid. The function k(α) : x ∈ Ω 7→

117

k(α(x)) is the dimensionless diffusivity defined as k(α)|

Ω_f

= 1 and k(α)|

Ω_s

= k

s

/k

f 118

with k

s

and k

f

are respectively the diffusivities of the solid and the fluid. We also

119

assume that k is a smooth regularization of (k

_s

/k

_f

)1

_Ωs

+ 1

_Ωf

. In this framework, α

120

is thus defined as a parameter function, which will let us control the distribution of

121

the solid in Ω.

122

Let us now specify the boundary conditions. Assume ∂Ω = Γ is Lipschitz and we

123

split it in three parts: Γ = Γ

_w

∪Γ

_in

∪Γ

_out

. Here, Γ

_w

are the walls, Γ

_in

the inlet/entrance

124

and Γ

out

is the exit/outlet of the computational domain. As exposed above, we would

125

like to rigorously study a non-smooth outlet boundary condition. Inspired by [13],

126

the following formulation tries to encapsulate these different approaches. Let β be

127

a function defined on Γ

out

and define: ∀x ∈ R : x

⁺

= pos(x) = max(0, x), x

⁻

=

128

neg(x) = max(0, −x), x = x

⁺

−x

⁻

. On top of (1.1), we impose the following boundary

129

conditions:

130

(1.2)

On Γ

in

: u = u

in

, θ = 0, On Γ

w

: u = 0, Ck∂

n

θ = φ,

On Γ

out

: A∂

n

u − np = A∂

n

u

^ref

− np

^ref

− 1

2 (u · n)

⁻

(u − u

^ref

), Ck∂

n

θ + β (u · n)

⁻

θ = 0,

131

with φ ∈ L

²

(0, T ; L

²

(Γ

w

)), f ∈ L

²

(0, T ; H

⁻¹

(Ω)), u

in

∈ L

²

(0, T ; H

₀₀^1/2

(Γ

in

)), n de-

132

notes the normal vector to the boundary, ∂

n

= n·∇ and (u

^ref

, p

^ref

) denotes a reference

133

solution. As stated in [29], this nonlinear condition is physically meaningful: if the

134

flow is outward, we impose the constraint coming from the selected reference flow ;

135

if it is inward, we need to control the increase of energy, so, according to Bernoulli’s

136

principle, we add a term that is quadratic with respect to velocity.

137

To define a weak formulation of (1.1)-(1.2), we introduce V

^u

= {u ∈ C

^∞

(Ω; R

^d

);

138

u

_Γin∪Γw

= 0}, and define V

^u

(resp H

^u

) as the closure of V

^u

in (H

¹

(Ω))

^d

(resp. in

139

(L

²

(Ω))

^d

). Similarly, we define V

^θ

= {θ ∈ C

^∞

(Ω; R ); θ

_Γin

= 0}, and V

^θ

and H

^θ

140

accordingly. A weak formulation of (1.1)-(1.2) then reads as:

141

Z

Ω

∂

t

θϕ − Z

Ω

θu · ∇ϕ + Z

Ω

Ck∇θ · ∇ϕ + Z

Γ

(θ(u · n) − Ck∂

n

θ)ϕ = 0,

142

for all ϕ ∈ H

^θ

. However, from (1.2), we have:

143

Z

Γ

(θ(u · n) − Ck∂

n

θ)ϕ = − Z

Γw

φϕ + Z

Γout

(u · n) + β(u · n)

⁻

θϕ

− Z

Γ_out

βθ(u · n)

⁻

+ Ck∂

_n

θ ϕ

= − Z

Γ_w

φϕ + Z

Γ_out

(u · n) + β(u · n)

⁻

θϕ.

144

Therefore:

145

(WF.1) Z

Ω

∂

_t

θϕ − Z

Ω

θu · ∇ϕ + Z

Ω

Ck∇θ · ∇ϕ + Z

Γout

(u · n) + β(u · n)

⁻

θϕ =

Z

Γw

146

φϕ.

Doing similar computations with the Navier-Stokes system yield:

147

(WF)

148 149

(WF.2)

Z

Ω

q∇ · u = 0, ∀q ∈ L

²

(Ω),

150 151

(WF.3)

Z

Ω

(∂

t

u + (u · ∇)u) · Ψ + A Z

Ω

∇u : ∇Ψ − Z

Ω

Bθe

y

· Ψ

− Z

Ω

p∇ · Ψ + Z

Ω

hu · Ψ + 1 2 Z

Γout

(u · n)

⁻

(u − u

^ref

) · Ψ

= Z

Ω

f · Ψ + Z

Γ_out

(A∂

_n

u

^ref

− np

^ref

) · Ψ,

152

for all Ψ ∈ H

^u

.

153

1.2. The topology optimization problem. A goal of this paper is to analyze

154

the next topology optimization problem

155

(OPT)

min J (α, u, θ, p) s.t.

( (u, θ, p) solution of (WF) parametrized by α, α ∈ U

ad

,

156

where J is a given cost function and, for some κ > 0, we set U

ad

= {α ∈ BV(Ω)

157

: 0 ≤ α(x) ≤ α

max

a.e. on Ω, , |Dα|(Ω) ≤ κ}. BV(Ω) stands for functions of bounded

158

4

variations, as exposed in [2]. We recall that the weak-* convergence in BV(Ω) is

159

defined as follows [2]: (α

ε

)

ε

⊂ BV(Ω) weakly-* converges to α ∈ BV(Ω) if (α

ε

)

160

strongly converges to α in L

¹

(Ω) and (Dα

ε

) weakly-* converges to Dα in Ω, meaning:

161

ε→+∞

lim Z

Ω

νdDα

ε

= Z

Ω

νdDα, ∀ν ∈ C

0

(Ω),

162

where C

0

(Ω) denotes the closure, in the sup norm, of the set of real continuous

163

functions with compact support over Ω. We choose U

ad

as a subset of BV(Ω) since

164

it is a nice way to approximate piecewise constant functions, which is close to the

165

desired solid distribution.

166

It is classical for these problems to compute first order optimality conditions

167

(see e.g. [33, 44]). This approach needs smoothness of the control-to-state mapping.

168

However, the presence of the non-differentiable function neg(x) = x

⁻

makes this

169

approach impossible. Therefore, we adopt a smoothing approach, as studied in [35,

170

45], and we approximate the neg function with a C

¹

positive approximation, denoted

171

neg

_ε

. We suppose this approximation satisfies the following assumptions:

172

(A1) ∀x ∈ R , neg

_ε

(x) ≥ neg(x).

173

(A2) ∀x ∈ R , 0 ≤ neg

⁰_ε

(x) ≤ 1.

174

(A3) neg

_ε

converges to neg uniformly over R .

175

(A4) for every δ > 0, the sequence (neg

⁰_ε

)

ε>0

converges uniformly to 1 on [δ, +∞)

176

and uniformly to 0 on (−∞, −δ] as ε → +∞.

177

As presented in [45], we may choose:

178

(1.3) neg

_ε

(x) =

( x

⁻

if |x| ≥

_2ε¹

,

1 2

1

2

− εx

3 3 2ε

+ x

if |x| <

_2ε¹

.

179

We thus redefine (WF) with an approximation of neg, which gives:

180

(WFe.1) Z

Ω

(∂

t

u

ε

+ (u

ε

· ∇)u

ε

) · Ψ + A Z

Ω

∇u

ε

: ∇Ψ − Z

Ω

Bθe

y

· Ψ +

Z

Ω

h(α

ε

)u

ε

· Ψ − Z

Ω

p

ε

∇ · Ψ + 1 2 Z

Γout

neg

_ε

(u

ε

· n) (u

ε

− u

^ref_ε

) · Ψ

= Z

Ω

f · Ψ − Z

Γ_out

(A∂

_n

u

^ref

+ np

^ref

) · Ψ,

181

182

(WFe.2)

Z

Ω

u

_ε

· ∇q = 0,

183 184

(WFe.3)

Z

Ω

∂

t

θ

ε

ϕ − Z

Ω

θ

ε

u

ε

· ∇ϕ + Z

Ω

Ck∇θ

ε

· ∇ϕ

+ Z

Γout

((u

ε

· n) + β neg

_ε

(u

ε

· n)) θ

ε

ϕ = Z

Γw

φϕ,

185

for all (Ψ, ϕ, q) ∈ H

^u

× H

^θ

× L

²

(Ω). We then define the approximate optimal control

186

problem:

187

(OPTe)

min J (α

ε

, u

ε

, θ

ε

, p

ε

) s.t.

( (u

_ε

, θ

_ε

, p

_ε

) solution of (WFe) parametrized by α

_ε

, α

ε

∈ U

ad

.

188

As it will be made clear later, the control-to-state mapping in (WFe) is smooth, which

189

will let us derive first order conditions.

190

1.3. Plan of the paper. The rest of this introduction is dedicated to the pre-

191

sentation of some notations used in this article and some important results of the

192

literature. The core of this paper is organized in two sections. First, we will prove the

193

existence of solutions to (WFe), which will let us prove, with a compactness argument,

194

the existence of solutions to (WF). We then focus on the two dimensional case, where

195

we prove uniqueness of the solutions along with stronger convergence results. This

196

is an extension of the work done by [13], where only the pressure and the velocity

197

where considered, and to [6, 15], where the steady case was studied in depth, but the

198

results concerning the unsteady case were obtained using restrictive assumptions. We

199

then study the approximate optimal control problem (OPTe), for which we will derive

200

first order conditions. We conclude this paper with the convergence of the optimality

201

conditions of (OPTe), which let us design first order conditions of (OPT).

202

Notations. We denote by a . b if there exists a constant C(Ω) > 0 depending

203

only on Ω such that a ≤ C(Ω)b. Denote:

204

• A : V

^u

→ (V

^u

)

⁰

defined by hAu, vi

_(Vu)⁰,V^u

= A R

Ω

∇u : ∇v,

205

• B : V

^u

× V

^u

→ (V

^u

)

⁰

defined by hB(u, v), wi

_(Vu)⁰,V^u

= R

Ω

(u · ∇)v · w,

206

• T : V

^θ

→ (V

^u

)

⁰

defined by hT θ, vi

(V^u)⁰,V^u

= R

Ω

Bθe

y

· v,

207

• P : L

²

(Ω) → (V

^u

)

⁰

defined by hP p, wi

(V^u)⁰,V^u

= R

Ω

p∇ · w,

208

• N : V

^u

×V

^u

→ (V

^u

)

⁰

defined by hN (u, v), wi

(V^u)⁰,V^u

= R

Γout

neg(u·n)(v·w),

209

• N

ε

: V

^u

×V

^u

→ (V

^u

)

⁰

given by hN

ε

(u, v)), wi

_(Vu)⁰,V^u

= R

Γ_out

neg

_ε

(u · n) (v ·

210

w).

211

• C(α) : V

^θ

→ (V

^θ

)

⁰

defined by hC(α)θ, ϕi

_(Vθ)⁰,V^θ

= R

Ω

Ck(α)∇θ · ∇ϕ,

212

• D : V

^u

× V

^θ

→ (V

^θ

)

⁰

defined by hD(u, θ), ϕi

_(Vθ)⁰,V^θ

= R

Ω

θu · ∇ϕ,

213

• M : V

^u

× V

^θ

→ (V

^θ

)

⁰

defined by hM(u, θ), ϕi

_(Vθ)⁰,V^θ

= R

Γ_out

((u · n)+

214

βneg(u · n))θϕ,

215

• M

ε

: V

^u

× V

^θ

→ (V

^θ

)

⁰

defined by hM(u, θ), ϕi

_(Vθ)⁰,V^θ

= R

Γout

((u · n)+

216

βneg

_ε

(u · n))θϕ,

217

By a slight abuse of notation, we will still denote by σ

^ref

the element of (V

^u

)

⁰

defined

218

by hσ

^ref

, wi

_(Vu)⁰,V^u

= R

Γ_out

(A∂

n

u

^ref

− p

^ref

n) · w, h(α) : V

^u

→ (V

^u

)

⁰

the function

219

defined by hh(α)u, vi

_(Vu)⁰,V^u

= R

Ω

h(α)u · v, and φ the element of (V

^θ

)

⁰

defined by

220

hφ, ϕi

_(Vθ)⁰,V^θ

= R

Γ_out

φϕ.

221

Results from the literature. We now recall two results from the literature that will

222

be heavily used throughout this paper.

223

Proposition 1.1. ([10, Proposition III.2.35]) Let Ω be a Lipschitz domain of

224

R

^d

with compact boundary. Let p ∈ [1, +∞] and q ∈ [p, p

^∗

], where p

^∗

is the critical

225

exponent associated with p, defined as:

226



 

 

1

p^∗

=

¹_p

−

¹_d

for p < d, p

^∗

∈ [1, +∞[ for p = d, p

^∗

= +∞ for p > d.

227

Then, there exists a positive constant C such that, for any u ∈ W

^1,p

(Ω):

228

kuk

L^q(Ω)

≤ Ckuk

¹⁺

d q−^d_p L^p(Ω)

kuk

d p−^d_q W^1,p(Ω)

.

229

6

Proposition 1.2. ([10, Theorem III.2.36]) Let Ω be a Lipschitz domain of R

^d

230

with compact boundary, and 1 < p < d. Then for any r ∈ h

p,

^p(d−1)_d−p

i

, there exists a

231

positive constant C such that, for any u ∈ W

^1,p

(Ω):

232

ku

∂Ω

k

_Lr(∂Ω)

≤ Ckuk

¹⁻

d p+^d−1_r L^p(Ω)

kuk

d p−^d−1_r W^1,p(Ω)

.

233

In the case p = d, the previous result holds true for any r ∈ [p, +∞[.

234

2. Existence of solutions. In this section, we will focus on proving the exis-

235

tence of solutions to (WFe) and prove their convergence toward the ones of (WF).

236

We make the following assumptions throughout this paper:

237

Assumptions 2.1. • The source term f ∈ L

²

(0, T ; H

⁻¹

(Ω)).

238

• (u

^ref

, p

^ref

) are such that:

239



 

 

 

 

 

 

 

 

u

^ref

∈ L

^r

(0, T ; (H

¹

(Ω))

^d

) ∩ L

^∞

(0, T ; (L

²

(Ω))

^d

) with r = 2 if d = 2 and r = 4 if d = 3,

∇ · u

^ref

= 0,

∂

t

u

^ref

∈ L

²

(0, T ; (L

²

(Ω))

^d

), u

^ref

= u

in

on Γ

in

.

240

• There exists k

_min

such that k(x) ≥ k

_min

> 0 and h(x) ≥ 0 for a.e. x ∈ Ω.

241

• The initial condition u

₀

(resp. θ

₀

) is a Fr´ echet-differentiable function from

242

U

_ad

to V

^u

(resp. V

^θ

). Furthermore, for all α ∈ U

_ad

, u

₀

(α)

_Γ

in

= u

_in

(0),

243

u

0

(α)

_Γ

w

= 0, ∇ · u

0

(α) = 0 and θ

0

(α)

_Γ

in

= 0.

244

• β ∈ L

^∞

(0, T ; L

^∞

(Γ

_out

)) such that β(t, x) ≥

¹₂

, for a.e. (t, x) ∈ [0, T ] × Γ

_out

.

245

2.1. Existence in dimension 2 or 3. In this part, we work with a fixed ε > 0

246

and a given α

_ε

in U

ad

.

247

In order to prove the existence of solutions to (WFe), we follow the classical Fadeo-

248

Galerkin method, as used in [13, 36, 47]. By construction, V

^u

and V

^θ

are separable.

249

Therefore, both admit a countable Hilbert basis (w

_k^u

)

_k

and (w

^θ_k

)

_k

. Let us construct

250

an approximate problem, which will converge to a solution of the original problem

251

(WFe). Denote by V

_n^u

(resp. V

_n^θ

) the space spanned by (w

^u_k

)

_k≤n

(resp. (w

_k^θ

)

_k≤n

).

252

We consider the following Galerkin approximated problem:

253

find t 7→ v

n

(t) ∈ V

_n^u

, t 7→ p

n

(t) ∈ L

²

(Ω) and t 7→ θ

n

(t) ∈ V

_n^θ

such that, defining

254

u

n

= v

n

+ u

^ref

, (u

n

, p

n

, θ

n

) satisfy (WFe) for all t ∈ [0, T ] and for all (Ψ, q, ϕ) ∈

255

V

_n^u

× L

²

(Ω) × V

_n^θ

.

256

As done in [47], we prove that such (u

n

, θ

n

, p

n

) exist. We now prove that these

257

solutions are bounded with respect to n and ε:

258

Proposition 2.2. There exist positive constants c

^θ₁

, c

^θ₂

, c

^v₁

and c

^v₂

, independent

259

of ε and n, such that:

260

(2.1) sup

[0,T]

kθ

_n

k

_L2(Ω)

≤ c

^θ₁

,

261 262

(2.2)

Z

T 0

k∇θ

n

k

²_L2(Ω)

≤ c

^θ₂

,

263

264

(2.3) sup

[0,T]

kv

n

k

_L2(Ω)

≤ c

^v₁

,

265 266

(2.4)

Z

T 0

k∇v

n

k

²_L2(Ω)

≤ c

^v₂

.

267

Proof. Taking ϕ

n

= θ

n

in (WFe.1) and integrating by part give:

268

d

dt kθ

n

k

²_L2(Ω)

− 1 2

Z

Γ_out

θ

²_n

(u

_n

· n) + Z

Ω

Ck|∇θ

n

|

²

+ Z

Γout

((u

n

· n) + βneg

_ε

(u

n

· n)) θ

_n²

= Z

Γw

φθ

n

.

269

Since β ≥

¹₂

and using assumption (A1), one has on Γ

out

:

270

((u

n

· n) + βneg

_ε

(u

n

· n)) θ

²_n

− 1

2 (u

n

· n)θ

²_n

≥ 1

2 ((u

n

· n) + neg

_ε

(u

n

· n)) θ

²_n

≥ 1

2 (u

n

· n)

⁺

θ

_n²

≥ 0.

271

Therefore:

_dt^d

kθ

_n

k

²_L2(Ω)

+ Ck

_min

k∇θ

_n

k

²_L2(Ω)

≤ kφk

_L2(Γ_w)

kθ

_n

k

_L2(Γ_w)

. Using continuity

272

of the trace operator and Young’s inequality, one proves that there exists a positive

273

constant C(Ω) such that, for any ν > 0:

274

d

dt kθ

n

k

²_L2(Ω)

+ Ck

_min

k∇θ

n

k

²_L2(Ω)

≤ 1

2ν kφk

²_L2(Γw)

+ C(Ω)ν

2 (kθ

n

k

²_L2(Ω)

+ k∇θ

n

k

²_L2(Ω)

).

275

Taking ν small enough, we are left with:

276

d

dt kθ

n

k

²_L2(Ω)

≤ 1

2ν kφk

²_L2(Γw)

+ C(Ω)ν

2 kθ

n

k

²_L2(Ω)

.

277

Integrating this equation and using Gronwall’s lemma then give (2.1) and (2.2).

278

Now, take Ψ

n

= v

n

in (WFe.3). After some calculations, one gets:

279

d

dt |v

n

|

²

+ A|∇v

n

|

²

+ 1 2 Z

Γ_out

neg

_ε

(u

_n

· n) |v

n

|

²

+ Z

Ω

h|v

n

|

²

= Z

Ω

f

θ

· v

n

− Z

Ω

∂

t

u

^ref

· v

n

− A Z

Ω

∇u

^ref

: ∇v

n

+ Z

Ω

(u

n

· ∇)v

n

· u

^ref

− Z

Ω

hu

^ref

· v

n

+ Z

Γ_out

(A∂

n

u

^ref

− np

^ref

)v

n 280

where f

θ

= f + Bθ

n

e

y

. First, using (2.2), one has kf

θ

k

(H^u)⁰

≤ kf k

_(Hu)⁰

+ Bc

^θ₁

.

281

Secondly, using (A1) gives that R

Γ_out

neg

_ε

(u

n

· n) |v

n

|

²

≥ 0 and following then the

282

same pattern of proof as in [13, Proposition 2], one proves (2.3) and (2.4).

283

Following [47, 10], we need to bound the fractional derivatives of the solution in

284

order to prove some convergence results. For any real-valued function f defined on

285

[0, T ], define by ˜ f the extension by 0 of f to the whole real line R , and by F ( ˜ f )

286

the Fourier transform of ˜ f , which we define as: F ( ˜ f )(τ) = R

R

f ˜ (t)e

^−itτ

dt. Using the

287

Hausdorff-Young inequality [10, Theorem II.5.20] we can prove the

288

Proposition 2.3. For all σ ∈ [0,

¹₆

), there exists a constant C(σ) > 0 indepen-

289

dent of ε and n such that:

290

(2.5)

Z

R

|τ|

^2σ

F

f θ

n

2 (L²(Ω))^d

≤ C(σ),

291 292

(2.6)

Z

R

|τ|

^2σ

k F ( u f

n

)k

²_L2(Ω)

≤ C(σ).

293

8

Proof. We emphasize that (2.6) is proved if (2.5) holds by using [10, Proposition

294

VII.1.3] by replacing f by f

θ

= f + Bθe

y

. The proof of (2.5) consists in adapting the

295

one of [10, Proposition VII.1.3] and is thus omitted.

296

Combining the two previous results, we can now prove the following existence

297

theorem for (WFe).

298

Theorem 2.4. For all (v

0

, θ

0

) ∈ H

^u

× H

^θ

and all T > 0, there exists v

ε

∈

299

L

^∞

(0, T ; H

^u

) ∩ L

²

(0, T ; V

^u

), θ

ε

∈ L

^∞

(0, T, H

^θ

) ∩ L

²

(0, T ; V

^θ

) and p

ε

∈ W

^−1,∞

(0, T ;

300

L

²

(Ω)) solution of (WFe) such that, defining u

0

= v

0

+ u

^ref

(0) and u

ε

= v

ε

+ u

^ref

,

301

one has for all (Ψ, ϕ) ∈ V

^u

× V

^θ

such that ∇ · Ψ = 0: R

Ω

u

ε

· Ψ (0) = R

Ω

u

0

· Ψ,

302

R

Ω

θ

ε

ϕ

(0) = R

Ω

θ

0

ϕ. Moreover, one has v

⁰_ε

=

^dv_dt^ε

∈ L

^4d

(0, T ; (V

^u

)

⁰

) and θ

⁰_ε

∈

303

L

²

(0, T ; (V

^θ

)

⁰

).

304

Proof. The proof of existence is similar to part (iv) of the proof of [47, Theorem

305

3.1] and the proof of [10, Proposition VII.1.4], where estimates (2.1)-(2.4) and (2.5)-

306

(2.6) are used in a compactness argument.

307

We only add the proof that (u

n

, θ

n

) converges to a solution of (WFe.1). Using

308

(2.5) and [47, Theorem 2.2], one shows that, up to a subsequence, θ

n

strongly con-

309

verges to an element θ

ε

of L

²

(0, T ; H

^θ

). The only technical points which needs more

310

detail are the non-linear terms in (WFe.1). Using the strong convergence of u

n

to u

ε 311

in L

²

(0, T ; H

^u

) proved in [47, Eq (3.41)], one proves that (θ

n

u

n

) strongly converges

312

to θ

ε

u

ε

in L

¹

(0, T ; L

²

(Ω)). Furthermore, notice that:

313

Z

T 0

k(u

n

· n)θ

n

k

⁴³

L⁴3(Γ)

≤ Z

T

0

ku

n

k

⁴³

L⁸3(Γ)

kθ

n

k

⁴³

L⁸3(Γ)

≤C Z

T

0

ku

_n

k

_L¹³_2(Ω)

kθ

_n

k

_L¹³_2(Ω)

ku

_n

k

_H1(Ω)

kθ

_n

k

_H1(Ω)

≤Cku

n

k

_L¹³∞(0,T;L²(Ω))

kθ

n

k

_L¹³∞(0,T;L²(Ω))

ku

n

k

_L2(0,T;H¹(Ω))

kθ

n

k

_L2(0,T;H¹(Ω))

.

314

This inequality together with (2.1)-(2.4) proves that ((u

n

· n)θ

n

)

n

is bounded in

315

L

⁴³

(0, T ; L

⁴³

(Γ)), which is reflexive. Therefore, it proves that, up to a subsequence,

316

there exists a weak limit κ

1

in L

⁴³

(0, T ; L

⁴³

(Γ)) of ((u

n

· n)θ

n

)

_n

. A simple adapta-

317

tion of the above reasoning proves that (neg

_ε

(u

n

· n) θ

n

)

_n

weakly converges to some

318

κ

₂

in L

⁴³

(0, T ; L

⁴³

(Γ)). Using the strong convergence of θ

_n

in L

²

(0, T ; L

²

(Ω)), [10,

319

Proposition II.2.12] implies that:

320

((u

n

· n) + βneg

_ε

(u

n

· n))θ

n

* ((u

ε

· n) + β neg

_ε

(u

ε

· n))θ

ε

in L

⁴³

(0, T ; L

¹

(Γ))

321

obtained using the continuity of x ∈ R 7→ neg

_ε

(x). By uniqueness of the limit in

322

the sense of distribution, we can identify κ

1

+ βκ

2

with ((u

ε

· n) + βneg

_ε

(u

ε

· n))θ

ε

.

323

Therefore, (u

ε

, θ

ε

) is a solution of (WF.1).

324

The convergence of the weak derivative with respect to time of v

_ε

in L

^4d

(0, T ;

325

(V

^u

)

⁰

) is proved in [10, Proposition V.1.3]. Concerning the weak derivative with

326

respect to time of θ

ε

, it follows immediately from the fact that differentiation with

327

respect to time is continuous in the sense of distribution. Existence of the pressure

328

p

ε

follows from [10, Chapter V].

329

We now use the existence of solutions to the approximate problem (WFe) to prove

330

existence of solutions to the limit problem (WF), along with the convergence of the

331

approximate solutions to the solutions of (WF).

332

Theorem 2.5. Let (α

ε

) ⊂ U

ad

and α ∈ U

ad

such that α

ε

* α

∗

in BV. Define by

333

(v

ε

, θ

ε

, p

ε

) a solution of (WFe) parametrized by α

ε

, and define u

ε

= v

ε

+ u

^ref

. Then,

334

there exists (v, θ, p) ∈ L

^∞

(0, T ; H

^u

) ∩ L

²

(0, T ; V

^u

) × L

^∞

(0, T, H

^θ

) ∩ L

²

(0, T ; V

^θ

) ×

335

W

^−1,∞

(0, T ; L

²

(Ω)) such that, defining u = v + u

^ref

, up to a subsequence, we have

336

• u

ε

*

∗

u in L

^∞

(0, T ; H

^u

) and θ

ε

* θ

∗

in L

^∞

(0, T ; H

^θ

),

337

• u

ε

* u in L

²

(0, T ; V

^u

) and in L

²

(0, T ; (L

⁶

(Ω))),

338

• θ

ε

* θ in L

²

(0, T ; V

^θ

) and in L

²

(0, T ; (L

⁶

(Ω))),

339

• u

ε

* u in L

⁴

(0, T ; (L

²

(Γ))

^d

) and θ

ε

* θ in L

⁴

(0, T ; (L

²

(Γ))),

340

• u

ε

− −−−− →

ε→+∞

u in L

²

(0, T ; (L

²

(Ω))

^d

) and θ

ε

− −−−− →

ε→+∞

θ in L

²

(0, T ; (L

²

(Ω))),

341

• u

ε

− −−−− →

ε→+∞

u in L

²

(0, T ; (L

²

(Γ))

^d

) and θ

ε

− −−−− →

ε→+∞

θ in L

²

(0, T ; (L

²

(Γ))),

342

• p

ε

* p in L

^d4

(0, T ; L

²

(Ω)).

343

Furthermore, (v, θ, p) is a solution to (WF) parametrized by α.

344

Proof. Using (2.1)-(2.4) and (2.5)-(2.6), we prove that there exists u and θ such

345

that all the convergences above are verified in the same manner as in [10, Propo-

346

sition VII.1.4]. Let us prove first that u is a solution of (WF.3) parametrized

347

by α and θ. With the same pattern of proof as in Theorem 2.4, one proves im-

348

mediately that (u

ε

· ∇)u

ε

* (u · ∇)u in L

¹

(0, T ; (L

¹

(Ω))

^d

), and (u

ε

· n)u

^ref

*

349

(u · n)u

^ref

in L

⁴

(0, T ; (L

⁴³

(Γ))

^d

). Regarding the penalization term:

350

kh(α

ε

)u

ε

− h(α)uk

²_L2(0,T;L²(Ω)^d)

. khk

²_∞

ku

ε

− uk

²_L2(0,T;L²(Ω)^d)

+ Z

T

0

Z

Ω

(h(α

ε

) − h(α))

²

|u|

²

.

351

Since α

_ε

→ α strongly in L

¹

(Ω), h(α

_ε

) → h(α) pointwise in Ω up to a subsequence

352

(which is not relabeled). Lebesgue dominated convergence theorem then implies:

353

h(α

_ε

)u

_ε

− −−−− →

ε→+∞

h(α)u in L

²

(0, T ; (L

²

(Ω))

^d

).

354

Concerning the boundary terms, we only consider the term with the approxima-

355

tion of the neg function. First, we claim that there exists γ such that neg

_ε

(u

ε

· n) (u

ε

+

356

u

^ref

) * γ in L

⁴³

(0, T ; L

⁴³

(Γ)

^d

). Notice that, for ε large enough and using the proper-

357

ties of the neg approximation, we have:

358

(2.7) Z

T

0

kneg

_ε

(u

ε

· n) (u

ε

+ u

^ref

)k

⁴³

L⁴3(Γ)

. Z

T

0

ku

ε

· n + 1k

⁴³

L⁸3(Γ)

ku

ε

+ u

^ref

k

⁴³

L⁸3(Γ)

. Z

T

0

ku

ε

k

⁴³

L⁸3(Γ)

+ C

ku

ε

k

⁴³

L⁸3(Γ)

+ Z

T

0

ku

ε

k

⁴³

L⁸3(Γ)

+ C

ku

^ref

k

⁴³

L⁸3(Γ)

. Z

T

0

ku

ε

k

⁸³

L⁸3(Γ)

+ 2 Z

T

0

ku

ε

k

⁸³

L⁸3(Γ)

!

¹₂

+ C.

359

In addition, from Proposition 1.2, we have ku

ε

k

⁸³

L⁸3(Γ)

. ku

ε

k

_L²³2(Ω)

ku

ε

k

H¹(Ω)

. Since

360

u

_ε

is bounded in L

^∞

(0, T ; (L

²

(Ω))

^d

) and in L

²

(0, T ; (H

¹

(Ω))

^d

) as proved in Propo-

361

sition 2.2, we see that neg

_ε

(u

ε

· n) (u

ε

+ u

^ref

) is bounded in L

⁴³

(0, T ; L

⁴³

(Γ)

^d

) in-

362

dependently of ε. Since this Banach space is reflexive, it proves the claimed weak

363

convergence. Let us now prove that γ can be identified with (u · n)

⁻

(u + u

^ref

).

364

10

First, since u

ε

→ u strongly in L

²

(0, T ; L

²

(Γ)

^d

) and neg

_ε

(·) → (·)

⁻

uniformly,

365

one proves that neg

_ε

(u

ε

· n) → (u · n)

⁻

strongly in L

²

(0, T ; L

²

(Γ)). Then, the

366

weak convergence of u

ε

in L

⁴

(0, T ; L

²

(Γ)

^d

) and [10, Proposition II.2.12] implies that

367

neg

_ε

(u

_ε

· n) (u

_ε

+ u

^ref

) * (u · n)

⁻

(u + u

^ref

) weakly in L

⁴³

(0, T ; L

¹

(Γ)

^d

). Using [10,

368

Proposition II.2.9], we argue that γ = (u · n)

⁻

(u + u

^ref

).

369

Regarding p

_ε

, we use an inf-sup condition as the one introduced in the proof of

370

[28, Theorem 5.1, eq. (5.14)], which states that

371

(2.8) kpk

_L2(Ω)

. sup

Ψ∈V

R

Ω

p∇ · Ψ kΨk

_H1(Ω)

.

372

Therefore, using (WFe.3), one shows that:

373

kp

ε

k

L²(Ω)

. k∂

t

u

_ε

k

V⁰

+ kB(u

ε

, u

_ε

)k

V⁰

+ kAu

ε

k

V⁰

+ kh(α)u

ε

k

V⁰

+ kT θk

V⁰

+ kN

ε

(u

ε

, u

ε

− u

^ref

)k

V⁰

+ kfk

V⁰

+ kσ

^ref

k

V⁰

.

374

We now bound each term depending on ε:

375

• Since the Stokes operator is continuous, kAu

ε

k

V⁰

. ku

ε

k

H¹(Ω)

and therefore,

376

Au

ε

is bounded in L

²

(0, T ; V

⁰

).

377

• Using [10, Eq. (V.3)], we prove that kB(u

ε

, u

ε

)k

V⁰

. ku

ε

k

²⁻_L2(Ω)^d2

ku

ε

k

_H^d21(Ω)

,

378

which in turn shows that B(u

ε

, u

ε

) is bounded in L

^4d

(0, T ; V

⁰

).

379

• Obviously, kh(α)u

ε

k

V⁰

≤ khk

_∞

ku

ε

k

_L2(Ω)

and therefore, h(α)u

ε

is bounded

380

in L

^∞

(0, T ; V

⁰

).

381

We are left with the boundary term. Let 0 6= Ψ ∈ V . In a similar manner as before

382

and using Proposition 1.2, there exists a constant C > 0 such that:

383

1 kΨk

_H1(Ω)

Z

Γout

|neg

_ε

(u

ε

· n) (u

ε

− u

^ref

) · Ψ| . ku

ε

k

3−d 4

L²(Ω)

ku

ε

k

d−1 4

H¹(Ω)

+ C

²

.

384

As proved before, u

_ε

is bounded in L

^∞

(0, T ; L

²

(Ω)) ∩ L

²

(0, T ; H

¹

(Ω)). Therefore, u

_ε

385

is also bounded in L

^2−2d

(0, T ; H

¹

(Ω)). Taking the supremum over Ψ, this proves that

386

N (u

_ε

, u

_ε

) is bounded in L

^4d

(0, T ; V

⁰

). Finally, in a similar fashion as in [10, Lemma

387

V.1.6], the above bounds prove that ∂

t

u

ε

is bounded in L

^4d

(0, T ; V

⁰

). These bounds

388

prove that (p

_ε

) is bounded in L

^4d

(0, T ; L

²

(Ω)), and therefore (p

_ε

) weakly converges to

389

some p in L

^d4

(0, T ; L

²

(Ω)).

390

Concerning θ, the convergence is largely proved in the same way as in Theorem 2.4.

391

The only difference concerns the convergence of neg

_ε

(u

ε

· n) θ

ε

to (u · n)

⁻

θ, which

392

is proved in the same manner as (2.7). All these convergence results let us say that

393

(u, θ, p) is a solution to (WF) in the distribution sense.

394

2.2. Further results in dimension 2. It is notably known that the solution

395

of the Navier-Stokes equations with homogeneous Dirichlet boundary conditions are

396

unique in dimension 2. We prove here that uniqueness still holds with the boundary

397

conditions (1.2). We only sketch the proof.

398

Proposition 2.6. Let d = 2. Then, the solution (u

ε

, θ

ε

, p

ε

) of (WFe) is unique.

399

Sketch of proof First of all, note that uniqueness of (u

ε

, θ

ε

) implies the uniqueness of

400

p

ε

via the De Rham Theorem [10, Theorem IV.2.4 and Chapter V].

401

Let (u

ε1

, θ

ε1

) and (u

ε2

, θ

ε2

) be two solutions of (WF.1)-(WF.3). Define u = v =

402

v

ε1

− v

ε2

and θ = θ

ε1

− θ

ε2

. Slightly adapting the proof in [10, Section VII.1.2.5],

403

one proves that:

404

(2.9) d

dt |v|

²_L2(Ω)

+ A|∇v|

²_L2(Ω)

. g

^v

(t)|v|

²_L2(Ω)

+ B|θ|

²_L2(Ω)

+ ν

^v

|∇v|

²_L2(Ω) 405

where ν

^v

is a positive constant and g

^v

is a function in L

¹

([0, T ]).

406

Testing the differential equation verified by θ with θ proves that:

407

d

dt |θ|

²_L2(Ω)

+ 2C Z

Ω

k|∇θ|

²

+ Z

Γ_out

θ

²

1

2 (u

_ε1

· n) + βneg

_ε

(u

_ε1

· n)

= − Z

Γout

β (neg

_ε

(u

_ε1

· n) − neg

_ε

(u

_ε2

· n)) + 1 2 (u · n)

θ

_ε2

θ.

408

With a similar proof as the one of Proposition 2.2, we can prove that, on Γ

out

,

409

θ

^{2 1}₂

(u

1

· n) + β neg

_ε

(u

1

· n)

≥ 0. Therefore, using (A3), one has:

410

(2.10) d

dt |θ|

²_L2(Ω)

+ 2C Z

Ω

k|∇θ|

²

.

|β|

L^∞(Γ_out)

+ 1 2

|u · n|

L³(Γ_out)

|θ

ε2

|

L³(Γ_out)

|θ|

L³(Γ_out)

.

411

Using Sobolev embeddings and Young inequality, we prove:

412

|β |

L^∞(Γ_out)

+ 1 2

|u · n|

L³(Γ_out)

|θ

ε2

|

L³(Γ_out)

|θ|

L³(Γ_out)

.

|β|

_L∞(Γ_out)

+ 1 2

³

|θ

ε2

|

_L2(Ω)

|∇θ

ε2

|

²_L2(Ω)

2(ν

^θ

)

³

(|u|

²_L2(Ω)

+ |θ|

²_L2(Ω)

) + (ν

^θ

)

³²

2

|∇u|

²_L2(Ω)

+ |∇θ|

²_L2(Ω)

,

413

where ν

^θ

is a positive constant. Therefore, summing (2.9) and (2.10) gives

_dt^d

(|u|

²_L2(Ω)

+

414

|θ|

²_L2(Ω)

) . max(g

^v₁

, g

^θ

)(|u|

²_L2(Ω)

+ |θ|

²_L2(Ω)

), with g

₁^v

and g

^θ

integrable. Therefore,

415

applying Gronwall’s lemma and noticing that |u(0)|

²_L2(Ω)

+ |θ(0)|

²_L2(Ω)

= 0, one shows

416

that u = 0 and θ = 0.

417

Note that we may also prove that, for d = 2, the solution (u, θ, p) of (WF) is

418

unique. We can also state stronger convergence (compared to the ones stated in

419

Theorem 2.5) in dimension 2. These results will be useful in the analysis of the

420

optimisation problems.

421

Corollary 2.7. Suppose d = 2. Under the assumptions of Theorem 2.5, u

ε

→

422

u strongly in L

^∞

(0, T ; L

²

(Ω)

²

), ∇u

ε

→ ∇u strongly in L

²

(0, T ; L

²

(Ω)

²

), θ

ε

→ θ

423

strongly in L

^∞

(0, T ; L

²

(Ω)), ∇θ

ε

→ ∇θ strongly in L

²

(0, T ; L

²

(Ω)) and p

ε

→ p

424

strongly in L

²

(0, T ; L

²

(Ω)).

425

Proof. Denote ¯ u = u− u

ε

, ¯ θ = θ −θ

ε

and ¯ p = p

ε

−p. The variational formulation

426

verified by (¯ u, θ, ¯ p) reads as: for all ¯ Ψ ∈ V

^u

:

427

(2.11a)

−hP p, ¯ Ψi =h∂

t

u ¯ + A¯ u + h(α)¯ u, Ψi

(V^u)⁰,V^u

+ h(h(α) − h(α

_ε

))u

_ε

, Ψi

(V^u)⁰,V^u

+ hB(u, u) − B(u

ε

, u

ε

), Ψi

_(Vu)⁰,V^u

+ hT θ, ¯ Ψi

_(Vu)⁰,V^u

1

2 hN (u, u − u

^ref

) − N

ε

(u

ε

, u

ε

− u

^ref

), Ψi

_(Vu)⁰,V^u

,

428

429

(2.11b) 0 = h∇ · u, qi ¯

_L2(Ω)

,

430

12