Global weak solutions for some Oldroyd models

Global weak solutions for some Oldroyd models

J. Differential Equations 254 (2013) 660–685 Contents lists available at SciVerse ScienceDirect Journal of Differential Equations www.elsevier.com/l...

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J. Differential Equations 254 (2013) 660–685

Contents lists available at SciVerse ScienceDirect

Journal of Differential Equations www.elsevier.com/locate/jde

Global weak solutions for some Oldroyd models Olfa Bejaoui 1 , Mohamed Majdoub ∗,2 Université de Tunis El Manar, Faculté des Sciences de Tunis, Département de Mathématiques, El Manar 2092, Tunisia

a r t i c l e

i n f o

Article history: Received 26 June 2012 Revised 1 September 2012 Available online 25 September 2012 MSC: 35-XX 76-XX 76A05 35D30 74G25

a b s t r a c t We investigate an evolutive system of non-linear partial differential equations derived from Oldroyd models on non-Newtonian flows. We prove global existence of weak solutions, in the case of a smooth bounded domain, for general initial data. The results hold true for the periodic case. © 2012 Elsevier Inc. All rights reserved.

Keywords: Oldroyd models Weak solutions Global existence Young measure Equi-integrability

Contents 1. 2. 3. 4.

* 1 2

Introduction . . . . . . . . . . . . . . . . . . . . Notations and functional spaces . . . . . Technical lemmas and statement of the Proofs of the main results . . . . . . . . . . 4.1. Galerkin approximation . . . . . . 4.2. Uniform estimates . . . . . . . . . . 4.3. Strong convergence of v N in L 2 4.4. Equi-integrability of τ N in L 2 . .

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Corresponding author. E-mail addresses: [email protected] (O. Bejaoui), [email protected] (M. Majdoub). O.B. is grateful to the Laboratory of PDE and Applications at the Faculty of Sciences of Tunis. M.M. is grateful to the Laboratory of PDE and Applications at the Faculty of Sciences of Tunis.

0022-0396/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jde.2012.09.010

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4.5. Strong convergence of τ N 4.6. Weak limits of non-linear Acknowledgment . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

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1. Introduction The Navier–Stokes equation is a model that describes the evolution of incompressible fluids with constant viscosity called Newtonian fluids. However, many experiments show that the viscosity of a fluid may vary with the pressure, see Andrade [2], see also the book by Bridgman [5]. Further details and references to more recent experimental studies can be found in the book by Szeri [35] and in the paper by Malek and Rajagopal [31]. There are also many experiments which show that the viscosity may depend on the symmetric part of the velocity gradient. We can refer to Schowalter [34], Huilgol [16]. Recently, Malek, Necas and Rajagopal [29], Bulicek, Majdoub and Malek [6] have established existence results concerning the flows of fluids called non-Newtonian fluids, whose viscosity depends on both the pressure and the symmetric part of the velocity gradient. The constitutive equation for such fluids is given by





2 

T = − p I + ν p ,  D ( v )

D ( v ),

(1.1)

where T is the Cauchy stress, p is the pressure, v is the velocity, D ( v ) is the symmetric part of the velocity gradient and ν is the viscosity function. Thus, the evolution of such fluids is governed by the equation

ρ (∂t v + v .∇ v ) = div T , where ρ is the density of the fluid. However, there are some fluids that do not obey to the constitutive equation (1.1) such as Blood. Yet, it was shown experimentally that Blood is a complex rheological mixture that exhibits shear thinning and elastic behavior, see Thurston [36]. The constitutive equation for such fluids is given by

T = −p I + S, where S is the extra-stress tensor which is related to the kinematic variables through

S + λ1

DS Dt

2   DDv = 2μ  D ( v ) D ( v ) + 2λ2 , Dt

μ is the viscosity function, λ1 > 0 and λ2 > 0 are viscoelastic constants. The symbol

D Dt

denotes the objective derivative of Oldroyd type defined by

DS Dt

= ∂t S + v .∇ S + S . w ( v ) − w ( v ) S ,

where w ( v ) is the antisymmetric part of the velocity gradient. The extra-stress tensor S is decomposed into the sum of its Newtonian part

τs = 2

λ2 D (v ) λ1

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and its viscoelastic part

τe . The constitutive equation for τe is given by  2  λ2    τe + λ1 = 2 μ D (v ) − D ( v ). Dt λ1 D τe



Hence, we get the generalized Oldroyd-B model given by

⎧ λ2 ⎪ ⎪ ⎨ ρ ∂t v + ρ v .∇ v − λ  v + ∇ p = div(τe ), 1   2  λ2  1 2 ⎪ ⎪   ⎩ ∂t τe + τe + v .∇ τe + τe . w ( v ) − w ( v )τe = μ D (v ) − D ( v ). λ1 λ1 λ1 Particularly, when μ is constant, we recover the Oldroyd-B fluid with constant viscosity, see [28]. Existence of local strong solutions to the Oldroyd-B model was proved by Guillopé and Saut in [14,15]. Fernandez-Cara, Guillen and Ortega [11–13] proved local well-posedness in Sobolev spaces. In the frame of critical Besov spaces, Chemin and Masmoudi had proved local and global well-posedness results in [7]. In addition, non-blow up criteria for Oldroyd-B model were given in [18] as well as in [24]. For the sake of completeness, we refer the reader to the most recent papers about Oldroyd models as [20,19,22,21,25]. Global existence for small data was proved in [23] and [26]. Considering general initial data, Lions and Masmoudi [28] had established results of global existence of weak solutions when the viscosity function μ is constant. Our aim in this paper is to generalize the results in [28]. More precisely, we will focus on the following system

(S)

   ⎧ ∂t v + v .∇ v − div f D ( v ) + ∇ p = div τ , in R+ × Ω, ⎪ ⎪ ⎪   ⎨ ∂t τ + v .∇ τ + τ . w ( v ) − w ( v ).τ + aτ = g D ( v ) , in R+ × Ω, ⎪ ⎪ div v = 0, in R+ × Ω, ⎪ ⎩ v (0, x) = v 0 (x), τ (0, x) = τ0 (x),

where Ω can be considered either the torus Tn , or a smooth bounded domain of Rn , n = 2, 3 and in this case ( S ) is supplemented by the Dirichlet homogeneous boundary condition. The function g ( D ( v )) is given by







 

2 ˜  D ( v ) D ( v ) = g D (v ) = μ r −2

b 1−θ

2     μ  D ( v ) − θ D ( v ),

where μ(| D ( v )|2 ) = 1 − λ + λ(1 + | D ( v )|2 ) 2 , λ ∈ [0, 1], r ∈ [1, 2]. 2(1−θ) 1 , b = We , where We is the Weissenberg Physically, the parameters a and b are given by a = We number and θ ∈ ]0, 1[ is the ratio between the so-called relaxation and times. ˜ is motivated by the system studied by Arada and SeLet us mention that the expression of μ queira [3] in the steady case with f being the identity map, where existence of a unique solution was established for small and suitably regular data. Notice that when the term f ( D ( v )) is replaced θ by ν  v, with ν = Re , Re being the Reynolds number, and r is equal to 2, then ( S ) turns into the system studied by Lions and Masmoudi in [28]. Let us remark that when b = 0 and τ0 = 0, then τ = 0 solves the second equation in ( S ) which consequently will be reduced to the system studied in [8] and [9]. Our objective is to prove global existence of weak solutions, for general initial data, under suitable hypotheses on the function f . The layout of this paper is as follows. In the following section, we give some notations and we introduce the functional spaces used along this paper. The third section is devoted to some technical lemmas and to the statement of the main results. In the fourth section, we prove existence

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663

of approximate solutions ( v N , τ N ) to the system ( S ) by using Galerkin method. Then, we prove the equi-integrability in the Lebesgue space L 2 ([0, T ] × Ω) of the sequence τ N , and we derive an evolution equation for η := |τ N − τ |2 . This enables us to get the strong convergence of the sequence τ N in L 2 ([0, T ] × Ω). Therefore, we can identify the weak limits in the sense of distributions of non-linear terms. 2. Notations and functional spaces Let us introduce some notations which we use throughout this paper. With M n×n we denote the n×n , and for a vector field v, real vector space of n × n matrices. We have (div τ )i = j ∂ j τi j , for τ ∈ M we have (∇ u )i j = ∂ j u i . 2 For A = (ai , j ) and B = (b i , j ) in M n×n , A : B stands for the sum i , j ai , j b i , j and | A | denotes the quantity A : A. Let us denote by

V = φ ∈ D(Ω)n , div φ = 0 , and

 V = M ∈ D(Ω)n×n , M t = M , when Ω is a bounded domain of Rn ,

   n , V = φ ∈ C∞ (Ω) , div φ = 0 , φ dx = 0 per Ω

and

   n×n  V = M ∈ C∞ , Mt = M , M dx = 0 per (Ω) Ω

when Ω is the torus T n ,

H = the closure of V in the L 2 (Ω)n -norm.  = the closure of V  in the L 2 (Ω)n×n -norm. H 2 Vq = closure of V in the L q (Ω)n -norm of gradients, q  1.  in the L q (Ω)n×n2 -norm of gradients, q  1.  Vq = closure of V s ,2 V s = the closure of V with respect to the W (Ω)n -norm, with s > 1 + n2 .  with respect to the W s,2 (Ω)n×n -norm, with s > 1 + n .  V s = the closure of V 2 The condition on s is due to the fact that: if v ∈ W s,2 (Ω)n , then ∇ v ∈ W s−1,2 (Ω)n and W (Ω) → L ∞ (Ω) if 12 − s−n 1 < 0. We denote by V s the dual space of V s and by  , s the duality between V s and V s . The scalar product in L 2 (Ω) will be denoted by (. , .). For s > 1 + n2 and q  2, we have the following inclusions 2

s−1,2

V s ⊂ Vq ⊂ H H ⊂ Vq ⊂ V s ,

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and

  H  ⊂  Vs ⊂ Vq ⊂ H Vq ⊂  V s . For a sequence f N in D  ([0, T ] × Ω), we denote by f N its weak limit in the sense of distributions. By C we denote any constant that may depend on |Ω| the n-dimensional Lebesgue measure of Ω , v 0 , τ0 and on T > 0, but not on N. Let us mention that subsequences will not be relabeled. 3. Technical lemmas and statement of the main results We give some technical lemmas needed for the proofs of our main results. One of the properties of the norm on Banach spaces is the lower semi-continuity given by the following lemma. For more details, we refer to [38]. Lemma 3.1. Let X be a Banach space equipped with a norm . . Let (x N ) be a sequence in X that converges weakly to some x in X , then (x N ) is bounded in X and we have

x  lim inf xN . N →+∞

Lemma 3.2 (Vitali’s Lemma). Let Ω be a bounded domain in Rn and f N : Ω → R be integrable for every N ∈ N. Assume that (i) lim N →+∞ f N ( y ) exists and is finite for almost all y ∈ Ω , (ii) for every ε > 0, there exists δ > 0 such that

 sup

N ∈N

   f N ( y ) dy < ε ,

∀ H ⊂ Ω, | H | < δ.

H

Then



 f N ( y ) dy =

lim

N →+∞

Ω

lim

N →+∞

f N ( y ) dy .

Ω

For the proof of Lemma 3.2, we can refer to [1]. Lemma 3.3 (Aubin–Lions Lemma). Let 1 < α , β < +∞ and T > 0. Let X be a Banach space, and let X 0 , X 1 be separable and reflexive Banach spaces such that X 0 is compactly embedded into X which is continuously embedded into X 1 , then









v ∈ L α [0, T ], X 0 ; ∂t v ∈ L β [0, T ], X 1







is compactly embedded into L α [0, T ], X .

For the proof of Lemma 3.3, we refer to [27]. A generalized form of this lemma for locally convex spaces and β = 1 can be found in [33]. The following lemma plays an important role in the theory of existence of solutions to ordinary differential equations. The proof of such lemma can be found in [37].

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665

Lemma 3.4 (Caratheodory Theorem). Let c : I δ ≡ [t 0 − δ, t 0 + δ] → Rn the system of ordinary differential equations

⎧ ⎨ d (E)







c (t ) = F t , c (t ) ,

dt c (t 0 ) = c 0 ∈ Rn ,

t ∈ Iδ ,

Assume F : I δ × K → Rn , where K ≡ {c ∈ Rn , |c − c 0 | < β}, for some β > 0. If F satisfies the Caratheodory conditions: (i) t → F i (t , c ) is measurable for all i = 1, . . . , n and for all c ∈ K , (ii) c →  F (t , c ) is continuous for almost all t ∈ I δ , (iii) there exists an integrable function G : I δ → R such that

   F i (t , c )  G (t ),

∀(t , c ) ∈ I δ × K , ∀i = 1, . . . , n,

then there exist δ  ∈ ]0, δ[ and a continuous function c : I δ  → Rn such that (i) dc exists for almost all t ∈ I δ  , dt (ii) c solves E. The following theorem, the proof of which can be found in [4], deals with compact injections of Sobolev spaces into Lebesgue ones. Theorem 3.1 (Rellich–Kondrachov). Let Ω be a C 1 bounded domain of Rn . We have the following compact injections (i) if p = n, then W 1, p (Ω) ⊂ L q (Ω), ∀q ∈ [1, +∞[, (ii) if p > n, then W 1, p (Ω) ⊂ C (Ω). In order to prove the existence of approximate solutions, we use Galerkin method by considering a special basis in the space V s (Ω) as in [8]. The proof of existence of such basis, see [30], relies on solving the following spectral problem: find w r ∈ V s and λr ∈ R satisfying



wr , ψ

 s

  = λr w r , ψ ,

∀ψ ∈ V s .

(P )

r ∞ Theorem 3.2. There exist a countable set {λr }r∞ =1 and a corresponding family of eigenvectors { w }r =1 solving the problem (P ) such that 

(i) ( w r , w r ) = δr ,r  , ∀r , r  ∈ N, (ii) 1  λ1  λ2  · · · and λr → ∞ as r → ∞, r

r

λr

λr 

w (iii)  √w , √

s = δr ,r  , ∀r , r  ∈ N,

(iv) { w r }r∞ =1 forms a basis in V s . Moreover defining H N := span{ w 1 , . . . , w N } (a linear hull) and

P N ( v ) :=

N   i =1



v , w r w r ∈ HN ,

v ∈ V s,

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we obtain

 N P 

L( V s , V s )

 1,

 N P 

L( V s , V s )

 N P 

 1,

L(H,H)

 1.

The same results hold if we replace V s by  V s. One of the tools that we use in order to prove our main results is Young measures. Thus, we should give some well-known facts about this tool. We can refer to [32] as well as to [17] and [30] for more details. Theorem 3.3. Let ( f N : Ω → Rn ) be a bounded sequence in L 1 (Ω) and f N be its weak limit in D  (Ω). Assume that ( f N ) is equi-integrable in L 1 (Ω), then

 fN =

λ dνx (λ),

a.e. x ∈ Ω,

Rn

where ν is the Young measure generated by the sequence f N . Now, assume that f N converges weakly in L 1 (Ω), then f N is equi-integrable in L 1 (Ω) and

 fN =

λ dνx (λ),

a.e. x ∈ Ω.

Rn

Theorem 3.4. Let ( f N : Ω → Rn ) be a sequence of maps that generates the Young measure ν . Let F : Ω × Rn → R be a Caratheodory function that is a function that satisfies the Caratheodory conditions. Assume that the negative part F − (., f N (.)) is weakly relatively compact in L 1 (Ω), then

 

 F (x, λ) dνx (λ) dx  lim inf

Ω Rn





F x, f N (x) dx.

N →+∞

Ω

Now, we are ready to state our main results. The first theorem gives a global existence result for the following system

(S1)

   ⎧ ∂t v + v .∇ v − div f D ( v ) + ∇ p = div τ , in R+ × Ω, ⎪ ⎪ ⎨   ∂t τ + v .∇ τ + aτ = g D ( v ) , in R+ × Ω, + ⎪ ⎪ ⎩ div v = 0, in R × Ω, v (0, x) = v 0 (x), τ (0, x) = τ0 (x).

Theorem 3.5. Let f : M n×n → M n×n be a continuous function satisfying the following hypotheses for some p ∈ ]2, +∞[ when n = 2 and p ∈ ] 52 , +∞[ when n = 3:

( H 1 ) growth: there exist c > 0 and c˜  0 such that

   f ( A )  c˜ + c | A | p −1 ,

∀ A ∈ M n×n ,

f (0) = 0,

( H 2 ) monotonicity: there exists ν > 0 such that









f ( A ) − f ( B ) : ( A − B )  ν | A − B |2 + | A − B | p ,

(Ω) respectively. Let v 0 and τ0 be in H(Ω) and H

∀ A , B ∈ M n×n .

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667

(i) If ν satisfies 2ν (1 − θ) > 1, then for an arbitrary λ ∈ [0, 1], there exists a global weak solution ( v , τ ) to the system ( S 1 ) such that









v ∈ L ∞ R + , H(Ω) ∩ L p 0, T , V p (Ω) ,





(Ω) , ∀ T > 0. τ ∈ L∞ R +, H

(ii) If ν is such that 0  2ν (1 − θ)  1, then the same result holds for an arbitrary λ ∈ [0,



2ν (1 − θ)[.

2

Let us assume the existence of a scalar function U ∈ C 2 (Rn ), called potential of f , such that for ×n and i , j , k, l ∈ {1, . . . , n} some p > 1, C 1 > 0, C 2 > 0 we have for all η, ξ ∈ M nsym

∂ U (η) = f i j (η), ∂ ηi j U (0) =

(3.2)

∂ U (0) = 0, ∂ ηi j

(3.3)

  p −2 2 ∂ 2 U (η) ξi j ξkl  C 1 1 + |η| |ξ | , ∂ ηi j ∂ ηkl  2   ∂ U (η)    p −2   .  ∂ η ∂ η   C 2 1 + |η | ij kl

(3.4)

(3.5)

As consequence of these assumptions, there exists C > 0 such that

     f ( A )  C 1 + | A | p −1 . Moreover, for p  2, there exists





ν > 0 such that 



f ( A ) − f ( B ) : ( A − B )  ν | A − B |2 + | A − B | p ,

∀ A , B ∈ M n×n .

Standard examples of functions f whose potentials satisfy these assumptions are



 p −2

f ( A) = 1 + | A|

A

and



f ( A ) = 1 + | A |2

 p−2 2

A.

For more details about the existence of a such potential U and consequences of properties (3.2)–(3.5) we refer to [30]. The second theorem concerns the following system

(S2)

⎧    ∂t v + v .∇ v − div f D ( v ) + ∇ p = div τ , ⎪ ⎪ ⎪ ⎨ ∂t τ + v .∇ τ + aτ = bD ( v ), in R+ × Ω, ⎪ ⎪ div v = 0, in R+ × Ω, ⎪ ⎩ v (0, x) = v 0 (x), τ (0, x) = τ0 (x).

in R+ × Ω,

Theorem 3.6. Let f : M n×n → M n×n be a C 1 -function satisfying the following hypotheses for some p ∈ ]2, +∞[ when n = 2 and p ∈ ] 52 , +∞[ when n = 3:

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( H 1 ) growth: there exists c > 0 such that

   f ( A )  c | A | p −1 ,

∀ A ∈ M n×n ,

( H 2 ) coercivity: there exists ν > 0 such that

∀ A ∈ M n×n ,

f ( A) : A  ν | A|p ,

( H 3 ) monotonicity





f ( A ) − f ( B ) : ( A − B )  0,

∀ A , B ∈ M n×n .

(Ω) respectively. Then, there exists a global weak solution ( v , τ ) to the Let v 0 and τ0 be in H(Ω) and H system ( S 2 ) such that









v ∈ L ∞ R + , H(Ω) ∩ L p 0, T , V p (Ω) ,





(Ω) , ∀ T > 0. τ ∈ L∞ R +, H

Obviously, if f satisfies the hypotheses of Theorem 3.5 with c˜ = 0, then f satisfies the hypotheses of Theorem 3.6. Let us remark that the quadratic term τ . w ( v N ) − w ( v N ).τ is not present in ( S 1 ) neither in ( S 2 ). The difficulty of this fact will be explained in the proofs of Theorems 3.5 and 3.6 that will be given in the case of a bounded domain. However, they can be easily adapted to the periodic case. 4. Proofs of the main results The results in Sections 4.1, 4.2, 4.3 and 4.4 hold for the system ( S ) and thus for the systems ( S 1 ) and ( S 2 ). 4.1. Galerkin approximation We will show the existence of approximate solution to the system ( S ) via Galerkin approximation r ∞  as in [8]. Hence, let {ar }r∞ =1 and {α }r =1 be bases of V s and V s respectively given by Theorem 3.2. Let T > 0 and N  1 be fixed. We define

v N (t , x) =

N  k =1

dkN (t )ak (x),

τ N (t , x) =

N 

ckN (t )αk (x),

k =1

where the coefficients ckN (t ) and dkN (t ) solve the so-called Galerkin system: a system of 2N non-linear equations with 2N unknowns

(S N )

   ⎧  N  N i   N  N i i ⎪ : ∂ v a dx + v .∇ v a dx + f D v Da dx = − τ N : Dai dx, ⎪ t ⎪ ⎪ ⎪ ⎪ Ω Ω Ω ⎪ ⎨Ω       N 2   N  j N j N N j N j ˜  D v  D v : α dx, ∂t τ : α dx + v .∇ τ : α dx + a τ : α dx = μ ⎪ ⎪ ⎪ ⎪ ⎪ Ω Ω Ω Ω ⎪ ⎪     ⎩ N c j (0) = τ0 , α j , diN (0) = v 0 , ai , 1  i , j  N .

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The initial conditions on ckN and dkN are such that

τ N (0, x) =  P N τ0 (x),

v N (0, x) = P N v 0 (x),

 respectively onto the linear where P N and  V N are the orthogonal continuous projectors of H and H hulls of the first eigenvectors ar , r = 1, . . . , N and α r , r = 1, . . . , N respectively. r ∞  The orthogonality of the two bases {ar }r∞ =1 and {α }r =1 in H and H respectively imply that the system ( S N ) can be rewritten as

⎧  ⎨ d c N , d N  = F t , c N , . . . , c N , d N , . . . , d N , i, j N N 1 1 dt j i     ⎩ N j N i c j (0) = τ0 , α , di (0) = v 0 , a , where for 1  i , j  N

  Fi , j t , c 1N , . . . , c NN , d1N , . . . , d N N    N   N    N   N k i N k N k ck α : Da dx − dk a . dk ∇ a ai dx = − k =1

Ω



 −

f

  N

−a

dkN D

 k

 i

: D a dx, −

a

k =1

Ω

Ω

+

N 

k =1

Ω



 j

: α dx +

k =1

  N Ω

α

k

dkN w

 k a

k =1

 .

ckN

.

N 

 ckN ∇

α

k

: α j dx

k =1

  N 2   N        N k  N k j μ˜  D dk a  D dk a : α dx   k =1

Ω N 

 dkN ak

k =1

Ω

 ckN

k =1

  N



α

k

j

: α dx −

k =1

k =1

  N Ω

 ckN

α

k

.

k =1

N 

dkN w

 k a

 : α j dx.

k =1

N Let R > 0 and K ⊂ R2N be the ball of center (c 1N (0), . . . , c N (0), d1N (0), . . . , d NN (0)) and of radius R. We 2N consider Fi , j : [0, T ] × K → R . ˜ and f lead to the continuity of Fi , j over [0, T ] × K . In addition, thanks to the The continuity of μ 1 < 0, we get continuous inclusion W s−1,2 (Ω) → L ∞ (Ω) if 12 − s− n

  N   N       k    N k i α  2 . ck α : Da dx  C ( R , N ) Dai  L ∞ (Ω)  L (Ω)   Ω

k =1

k =1

In the same way, we have

  N   N  N       k       N k N k i a  2 .∇ ak  ∞ . dk a . dk ∇ a a dx  C ( R , N )ai  L 2 (Ω)  L (Ω) L (Ω)    Ω

k =1

k =1

From the growth hypothesis on f , we deduce that

k,k =1

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  N         i        N k i   ˜ dk D a : D a dx  c D a L 1 (Ω) + c  f   k =1

Ω

 N  p −1     i       D a  dx dkN D ak    

Ω



k =1

N    k  D a  ∞  C (R, N) L (Ω)

 p −1

  i  D a 

k =1

L ∞ (Ω)

   + c˜  D ai L 1 (Ω) . ˜ is bounded, we estimate in the same way the remaining terms in Fi , j to get As μ

   Fi , j t , c N , . . . , c N , d N , . . . , d N   C ( R , N ), N N 1 1 where C ( R , N ) is a constant that does not depend on t. N The standard Caratheodory theory provides the existence of continuous functions (c 1N , . . . , c N ,

d N N N N d1N , . . . , d N N ) solutions to ( S N ) at least for a short time interval with dt (c 1 , . . . , c N , d1 , . . . , d N ) is defined almost everywhere. The uniform estimates that we will derive in the next subsection enable us to extend the solution onto the whole time interval [0, T ].

4.2. Uniform estimates Multiplying the first equation in ( S N ) by diN (t ), then taking the sum over i = 1, . . . , N, we obtain

1 d 2 dt

  v N (t )22 L



 

f D vN

+ (Ω)



  : D v N dx = −

Ω







τ N : D v N dx. Ω

The monotonicity hypothesis on f leads to

1 d 2 dt

  v N (t )22

L (Ω)

   p + ν  D v N (t )  p

L (Ω)

  2 +  D v N (t )  2

L (Ω)









τ N : D v N dx.

−

(4.6)

Ω

Multiplying the second equation in ( S N ) by c Nj (t ), then taking the sum over j = 1, . . . , N, we obtain

1 d 2b dt

 τ N (t )22 L

2 a + τ N (t )L 2 (Ω) = (Ω) b

1 1−θ

Notice that (τ N . w ( v N ) − w ( v N ).τ N ) : τ N = 0 since



   N 2     μ D v  − θ τ N : D v N .

Ω

τ N is symmetric. Hence, we get

    2  p 1 d 1 d a  v N (t )22 τ N (t )22 + ν  D v N (t )  L p (Ω) + + τ N (t ) L 2 (Ω) L (Ω) L (Ω) 2 dt 2b dt b     N 2     1  μ  D v  − 1 τ N : D v N dx 1−θ Ω

    λ  τ N (t ) 2  D v N (t )  2  L (Ω) L (Ω) 1−θ     λ b √   N  a N  τ (t ) 2 .  ν  D v (t ) L 2 (Ω) . L (Ω) 1 − θ aν b

O. Bejaoui, M. Majdoub / J. Differential Equations 254 (2013) 660–685

671

Young inequality implies that

1 d 2 dt

 Let

γ :=

λ

  v N (t )22

L (Ω)

λ 2(1 − θ) 

2(1−θ)

b . aν



  2  p 1 d  N 2 a τ (t ) 2 + ν  D v N (t )  L p (Ω) + + τ N (t )L 2 (Ω) L (Ω) 

b aν

2b dt

 









b

a 2 2 ν  D v N (t ) L 2 (Ω) + τ N (t )L 2 (Ω) . b

Notice that in both cases (i) and (ii) in Theorem 3.5, we have

(4.7) (4.8)

γ < 1. No-

tice also that to absorb the terms at the right-hand side of (4.8), we do not need that f satisfies the strong monotonicity condition. However, we need just that f satisfies the coercivity condition f ( A ) : A  ν | A | p . In fact, as Ω is a bounded domain and p > 2, then thanks to Holder inequality and Young one

  N 2  D v (t )  2 L



  N 2  D v (t )  dx

= (Ω) Ω

   C |Ω|



 2p   N 2  2p  D v (t )  dx

Ω

 2   p  C |Ω| +  D v N (t )  L p (Ω) 

p

    p  C |Ω| +  D v N (t )  L p (Ω) . 

Finally, we obtain

1 d 2 dt

  v N (t )22

L (Ω)

   p 1 d  N 2 τ (t ) 2 + ν (1 − γ ) D v N (t )  L p (Ω) + L (Ω) 2b dt

 2 a + (1 − γ )τ N (t ) L 2 (Ω)  C .

(4.9) (4.10)

b

Particularly, this leads to the fact that

 N   c , . . . , c N , d N , . . . , d N 2  C , N

1

1

N

where C is a constant that does not depend on t neither on N. N This uniform boundedness with the continuity of (c 1N , . . . , c N , d1N , . . . , d NN ), we deduce that the N functions (c 1N , . . . , c N , d1N , . . . , d NN ) are defined on the whole interval [0, T ], see Zeidler [39] for more details. By Korn inequality, we have

 N  ∇ v (t )

L p (Ω)

    C  D v N (t )  L p (Ω) ,

and we deduce that we can extract subsequences such that

v N ∗ v

weakly in L ∞ (0, T , H),

vN  v

weakly in L p (0, T , V p ),

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O. Bejaoui, M. Majdoub / J. Differential Equations 254 (2013) 660–685

), τ N ∗ τ weakly in L ∞ (0, T , H ). τ N  τ weakly in L 2 (0, T , H The growth condition on f and the uniform boundedness of the sequence D ( v N ) in L p ([0, T ] × Ω)  imply that f ( D ( v N )) is bounded in L p ([0, T ] × Ω), and thus

 



f D vN

      f D vN weakly in L p [0, T ] × Ω .

In addition, as g ( D ( v N )) is bounded in L p ([0, T ] × Ω), then

 

g D vN



      g D vN weakly in L p [0, T ] × Ω .

Now, we want to derive some bounds on the pressure. Let us start by the two-dimensional case. By the Gagliardo–Nirenberg inequality, we have

 N v 

L 4 (Ω)

 1  1  C  v N  L22 (Ω) ∇ v N  L22 (Ω) .

Thus, by Holder inequality, we get

 N 4 v  4

L ([0, T ]×Ω)

 2  2  C  v N  L ∞ (0,T , L 2 (Ω)) ∇ v N  L 2 ([0,T ]×Ω) . 

As p  < 2, then v N ⊗ v N is bounded in L p ([0, T ] × Ω). Since v N is divergence free, we get

      p N = div div f D v N − v N ⊗ v N + τ N . 

The uniform boundedness of τ N and f ( D ( v N )) in L p ([0, T ] × Ω) imply that p N is also bounded in  L p ([0, T ] × Ω). Hence, we infer that 

pN  p





weakly in L p [0, T ] × Ω .

In the three-dimensional case, the Gagliardo–Nirenberg inequality implies

 N v 

L 4 (Ω)

 1  3  C  v N  L42 (Ω) ∇ v N  L42 (Ω) . 

Since p  52 , then Holder inequality leads to the fact that v N ⊗ v N is bounded in L p ([0, T ] × Ω). As above, we get 

pN  p





weakly in L p [0, T ] × Ω .

Let us remark that at this stage, in the three-dimensional case, we do not really need to have p  52 . 4



We need only to have p > 2 which lead to the fact that v N ⊗ v N is bounded in L 3 (0, T , L p (Ω)) and thus p N is bounded in L

min( p  , 43 )



(0, T , L p (Ω)).

O. Bejaoui, M. Majdoub / J. Differential Equations 254 (2013) 660–685

673

4.3. Strong convergence of v N in L 2 We would like to prove that ( v , τ , p ) is still a solution to ( S ). The difficulties appear when passing to the limit in the non-linear terms

 

v N .∇ v N ; v N .∇ τ N ; f D v N



       ; g D v N , τ N . w v N − w v N .τ N .

For the first two terms, we only need to prove that v N converges strongly to v in L 2 ([0, T ] × Ω). The  bounds on the pressure and on the term v N ⊗ v N imply that ∂t v N is bounded in L r (0, T , W −1, p (Ω)), N 2 for some r ∈ ]1, +∞[. In addition, v is bounded in L (0, T , V2 (Ω)). As V2 (Ω) is compactly embed ded in L 2 (Ω) which is continuously embedded in W −1, p (Ω), thanks to the Aubin–Lions Lemma, up to a subsequence

vN → v





strongly in L 2 [0, T ] × Ω .

Hence, we get





vN ⊗ vN → v ⊗ v

strongly in L 1 [0, T ] × Ω .

Since v N is divergence free, then

v N .∇ τ N =

n 

v iN ∂i τ N =

i =1

n    ∂i v iN τ N = div( v τ ). i =1

As v N converges strongly to v in L 2 ([0, T ] × Ω) and deduce that

τ N converges weakly to τ in L 2 ([0, T ] × Ω), we 

v iN τ N  v i τ



weakly in L 1 [0, T ] × Ω .

Thus, it remains to prove that

 

f D vN



  = f D (v ) ,

 

g D vN



  = g D (v )

and









τ N . w v N − w v N .τ N = τ . w ( v ) − w ( v ).τ . If we prove that τ N converges strongly to τ in L 2 ([0, T ] × Ω), then our objective will be achieved. To get the strong convergence of τ N , we will prove first that τ N is equi-integrable in L 2 uniformly in t ∈ [0, T ]:

 lim sup sup

M →+∞ n t ∈[0, T ]

Ω

 N 2 τ  χ

{|τ N | M } dx = 0.

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O. Bejaoui, M. Majdoub / J. Differential Equations 254 (2013) 660–685

4.4. Equi-integrability of τ N in L 2 We will show the equi-integrability of τ N in L 2 uniformly in t ∈ [0, T ]. The idea is inspired from [28]. Let R > 0 be fixed. We decompose τ N into the sum of ψ N and H N that solve respectively the following systems



E 1N





       ∂t ψ N + v N .∇ψ N + aψ N + ψ N . w v N − w v N .ψ N = g D v N , ψ N (0, x) = τ0N (x)χ{|τ N |< R } , 0

and



E 2N





    ∂t H N + v N .∇ H N + aH N + H N . w v N − w v N . H N = 0, H N (0, x) = τ0N (x)χ{|τ N | R } . 0

Let M > 0, we have



 N 2 τ  χ



 N  ψ + H N 2 χ N {|τ | M } dx

{|τ N | M } dx =

Ω

Ω



2

 N 2 ψ  χ

 {|τ N | M }

dx + 2

Ω

 N 2 H  χ

{|τ N | M } dx.

Ω

Energy estimates on H N in the energy space L 2 (Ω) imply that



 N   H (t , x)2 dx 

Ω



 N    H (0, x)2 dx = τ N χ 0

2 

{|τ0N | R } L 2 (Ω) .

Ω

Noticing that ψ N (0, .) ∈ L p (Ω), we should have a bound on ψ N in L ∞ (0, T , L p (Ω)). We have

1 2

 2  2  2 1    ∂t ψ N  + aψ N  + v N .∇ ψ N  = g D v N : ψ N . 2

Multiplying this equation by p |ψ N | p −2 , we get

 p  p   p −2   N   p : ψN. ∂t ψ N  + ap ψ N  + v N .∇ ψ N  = p ψ N  g D v ˜ is bounded, integrating over Ω , we deduce that As μ

 ∂t

  N ψ (t , x) p dx  p

Ω



   N    ψ (t , x) p −1  D v N (t , x) dx.

Ω

Thanks to Holder inequality, we get

 ∂t Ω

  N ψ (t , x) p dx  p

 Ω

 p−p 1    1p    N   N ψ (t , x) p dx  D v (t , x) p dx . Ω

O. Bejaoui, M. Majdoub / J. Differential Equations 254 (2013) 660–685



1

Multiplying this equation by ( Ω |ψ N (t , x)| p dx) p

  ∂t ψ N (t , .) L p (Ω) 

−1



675

, we obtain

 1p   N   D v (t , x) p dx ,

Ω

and thus by Holder inequality in time

 N    1   ψ (t , .) p  ψ N (0, .) L p (Ω) + T 1− p  D v N  L p ([0,T ]×Ω) . L (Ω) As D ( v N ) is uniformly bounded in L p ([0, T ] × Ω), we get

 N  ψ (t , .)

L p (Ω)

   ψ N (0, .) L p (Ω) + C .

Applying Holder inequality, we obtain



 N 2 ψ  χ

 {|τ N | M }

dx 

Ω

 2p   1− 2p  N  ψ (t , x) p dx χ{|τ N |M } dx

Ω

Ω

 2  ψ N L ∞ (0,T , L p (Ω))





1 M2

1− 2p  N  τ (t , x)2 dx

Ω

2   C ψ N (0, .) p

L (Ω)

Since

+C



1 M

2(1− 2p )

 N 2(1− 2p ) τ  ∞ L

(0, T , L 2 (Ω))

τ N is uniformly bounded in L ∞ (0, T , L 2 (Ω)), then we obtain 

 N 2 ψ  χ N {|τ | M } dx  C

Ω

  N  ψ (0, .)2 p +C . L (Ω)

1 M

2(1− 2p )

But, we have as p > 2

 N  ψ (0, .) pp = L (Ω)



 N  τ (0, x) p χ

{|τ0N |< R } dx

Ω





 N    τ (0, x) p −2 τ N (0, x)2 χ

{|τ0N |< R } dx

Ω

 2  R p −2 τ N (0, .) L 2 (Ω) . Since, we have P N L(H  ,H )  1, we deduce that

 N  ψ (0, .)

L p (Ω)

R

p −2 p

 N 2 τ (0, .) p2 L

R (Ω)

p −2 p

2

τ0 Lp2 (Ω) .

.

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O. Bejaoui, M. Majdoub / J. Differential Equations 254 (2013) 660–685

Finally, for a fixed R > 0, we get

 sup sup

N t ∈[0, T ]

 N 2 τ  χ

{|τ N | M } dx  2 sup N

Ω

 N τ χ 0

2 

{|τ0N | R } L 2 (Ω)



1

+C M

2(1− 2p )

R

2( p −2) p

4  τ0 Lp2 (Ω) + C .

As p > 2, we obtain

 lim sup sup sup M

N t ∈[0, T ]

 N 2 τ  χ

{|τ N | M } dx  2 sup N

Ω

 N τ χ 0

2 

{|τ0N | R } L 2 (Ω) ,

(4.11)

which hold true for every R ∈ ]0, +∞[. The fact that τ0N converges strongly to τ0 in L 2 (Ω) implies that |τ0N |2 converges strongly in L 1 (Ω) and thus weakly in L 1 (Ω), and thus |τ0N |2 is equi-integrable in L 1 (Ω). Consequently, let R → +∞ in inequality (4.11), we infer that

 0  lim inf sup sup M

This means that

N t ∈[0, T ]

 N 2 τ  χ

 {|τ N | M }

dx  lim sup sup sup

Ω

M

N t ∈[0, T ]

 N 2 τ  χ

{|τ N | M } dx  0.

Ω

τ N is equi-integrable in L 2 (Ω) uniformly in t ∈ [0, T ].

4.5. Strong convergence of τ N in L 2 We will first focus on the system ( S ). As τ N is equi-integrable in L 2 (Ω) uniformly in t ∈ [0, T ], then |τ N − τ |2 converges weakly in 1 L ([0, T ] × Ω). So let η := |τ N − τ |2 ∈ L ∞ (0, T , L 1 (Ω)). Notice that if η = 0, then

 N    τ − τ 2 → 0 strongly in L 1 [0, T ] × Ω . Hence, our aim will be to show that η = 0. Multiplying by τ N the equation satisfied by

τ N , we get

 2     2 ∂t + v N .∇ τ N  + 2aτ N  = 2g D v N : τ N .



(4.12)

Notice here that the term v N .∇ τ N : τ N has a sense since ∇ τ N ∈ L ∞ , v N ∈ L 2 , τ N ∈ L 2 for a fixed N ∈ N. Let us introduce the unique a.e. flow in the sense of DiPerna and Lions [10] of v N , solution of

  ∂t X N (t , x) = v N t , X (t , x) ,

X N (0, x) = x.

We also denote by X the a.e. flow of v. Eq. (4.12) can be written as

 2   2          ∂t τ N  t , X N (t , x) + 2a τ N  t , X N (t , x) = 2 g D v N : τ N t , X N (t , x) . Passing to the limit weakly, we get

 2   2          ∂t τ N  t , X N (t , x) + 2aτ N  t , X N (t , x) = 2 g D v N : τ N t , X N (t , x) .

O. Bejaoui, M. Majdoub / J. Differential Equations 254 (2013) 660–685

677

The stability of the notion of a.e. flow with respect to the weak limit of v N to v implies that X N (t , x) converges to X (t , x) in L 1loc and also that ( X N (t )−1 )(x) converges to ( X (t )−1 )(x) in L 1loc . Thus, we get

 2          τ N  t , X N (t , x) = τ N 2 t , X (t , x) = |τ |2 + η t , X (t , x) . Finally, we obtain

1 2

       (∂t + v .∇) |τ |2 + η + a |τ |2 + η = g D v N : τ N .

(4.13)

In addition, since v N .∇ τ N converges in D  ([0, T ] × Ω) to v .∇ τ , the equation satisfied by

τ is then

       ∂t τ + aτ + v .∇ τ + τ N . w v N − w v N .τ N = g D v N , which implies that

1 2

       (∂t + v .∇)|τ |2 + a|τ |2 + τ N . w v N − w v N .τ N : τ = g D v N : τ .

Subtracting (4.14) from (4.13) gives the equation satisfied by

(4.14)

η

    ∂t η + 2aη + div(η v ) − 2τ N . w v N − w v N .τ N : τ         = 2 g D vN : τ N − g D vN : τ .

(4.15)

Notice that the term η v makes sense in the sense of distributions in the two-dimensional case as W 1, p (Ω) ⊂ L ∞ (Ω) and in the three-dimensional case for p > 3, and also the term τ N . w ( v N ) − w ( v N ).τ N : τ is not defined in the sense of distributions. To overcome this difficulty for any p, we use a renormalized form of (4.15) by multiplying such equation by (1+1η)2 , namely

(∂t + v .∇)ζ +

2a 1+η

ζ=

       2 g D vN : τ N − g D vN : τ (1 + η)2      + τ N . w v N − w v N .τ N : τ ,

where ζ = 1+η ∈ L ∞ ([0, T ] × Ω). Let us remark that η

 







T 1 := g D v N

           : τ N − g D v N : τ = g D v N − g D (v ) : τ N − τ ,

and





T 2 := τ N . w v N − w v N .τ N : τ =



 





 







τ N − τ . w v N − w (v ) − w v N − w (v ) . τ N − τ : τ .

τ N is equi-integrable in L 2 ([0, T ] × Ω) and w ( v N ) is bounded in L 2 ([0, T ] × Ω), then (τ − τ ).( w ( v N ) − w ( v )) is equi-integrable in L 1 ([0, T ] × Ω) and thus converges weakly in L 1 ([0, T ] × Ω). Let νt ,x be the Young measure generated by (τ N , w ( v N )), then we have As

N



 







τ N − τ . w v N − w (v ) = M n×n × M n×n

  (α − τ ). β − w ( v ) dνt ,x (α , β).

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O. Bejaoui, M. Majdoub / J. Differential Equations 254 (2013) 660–685

By Cauchy–Schwartz inequality, we get

      τ N − τ . w v N − w (v )  



 12 |α − τ | dνt ,x (α , β)



2

M n×n × M n×n





×

 12 |β − w ( v )|2 dνt ,x (α , β) .

M n×n × M n×n

The equi-integrability of

τ N in L 2 ([0, T ] × Ω) implies that  |α − τ |2 dνt ,x (α , β).

η= M n×n × M n×n

p

In addition, | w ( v N )|2 is equi-integrable in L 1 ([0, T ] × Ω) since it is bounded in L 2 ([0, T ] × Ω) with p > 2, thus

     w v N − w ( v )2 =



  β − w ( v )2 dνt ,x (α , β).

M n×n × M n×n

Consequently, we obtain

          τ N − τ . w v N − w ( v )   √η w v N − w ( v )2 12 , and we remark that

√     2 1      η  τ N − τ . w v N − w (v ) : τ   |τ | w v N − w ( v ) 2 , 2 2 (1 + η) (1 + η) 1



η

1

with (1+η)2 ∈ L ∞ ([0, T ] × Ω), |τ | ∈ L 2 ([0, T ] × Ω) and | w ( v N ) − w ( v )|2 2 ∈ L 2 ([0, T ] × Ω). As τ N is equi-integrable in L 2 ([0, T ] × Ω) and D ( v N ) is bounded in L 2 ([0, T ] × Ω), then (τ N − τ ).( g ( D ( v N )) − g ( D ( v ))) is equi-integrable in L 1 ([0, T ] × Ω). Let ν˜ t ,x be the Young measure generated by (τ N , D ( v N )), then we have



   (α − τ ). g (β) − g D ( v ) dν˜ t ,x (α , β).

T1 = M n×n × M n×n

As the derivative of g is bounded, then g is Lipschitz and the same calculus as for the term T 2 imply that

T1  C

2 1 √   N  η D v − D ( v ) 2 .

Finally, we get



(∂t + v .∇)ζ  C

   2 1 2 1  η   N  D v − D ( v ) 2 + |τ | w v N − w ( v ) 2 . 2 (1 + η)

O. Bejaoui, M. Majdoub / J. Differential Equations 254 (2013) 660–685

679

Now, we should estimate | D ( v N ) − D ( v )|2 and | w ( v N ) − w ( v )|2 in terms of first equation in ( S ). More precisely, we will prove that

  

f D vN



η with the help of the

           − f D (v ) : D (v ) − D v N = τ N − τ : D v N − D (v ) .

Multiplying the first equation in ( S ) by v N and using the identity

A ∈ M n×n , u ∈ Rn ,

div( A )u = div( Au ) − A : D (u ), we get

1 2

 2             ∂t  v N  − div f D v N v N + f D v N : D v N + v N .∇ v N . v N + div p N v N     = div τ N v N − τ N : D v N .

(4.16)

The strong convergence of v N to v in L 2 ([0, T ] × Ω) implies that | v N |2 converges strongly in L 1 ([0, T ] × Ω) to | v |2 . In addition, thanks to the weak convergence of τ N to τ in L 2 ([0, T ] × Ω), we can conclude that





τ N v N  τ v weakly in L 1 [0, T ] × Ω . 



But, v N is uniformly bounded in L p (0, T , V p ) and ∂t v N is uniformly bounded in L p (0, T , W −1, p (Ω)) in the two-dimensional case and L

min ( p  , 4 ) 3

(0, T , W

−1, p 

(Ω)) in the three-dimensional case. Since V p 

is compactly embedded into L p which is continuously embedded into W −1, p (Ω), Rellich–Kondrachov Theorem implies that





vN → v

strongly in L p [0, T ] × Ω . 

As f ( D ( v n )) is uniformly bounded in L p ([0, T ] × Ω), we obtain

 

f D vn



 

v N  f D vn



v





weakly in L 1 [0, T ] × Ω . 

In the two-dimensional case, p N and v N ⊗ v N are uniformly bounded in L p ([0, T ] × Ω). Thus, one gets

pN v N  pv





weakly in L 1 [0, T ] × Ω .

Notice that thanks to the divergence free condition on v N ,

v N .∇ v N . v N =

1 2







2 div  v N  v N ,

and thus

 N 2 N      v  v   v N 2 v = | v |2 v weakly in L 1 [0, T ] × Ω . 

In the three-dimensional case, for p  52 , p N and v N ⊗ v N are uniformly bounded in L p ([0, T ] × Ω). Notice that at this stage appears the bound on p when the dimension is three. In fact, we have

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O. Bejaoui, M. Majdoub / J. Differential Equations 254 (2013) 660–685

 N    2p   v ⊗ v N  pp   C  v N  L 4 (Ω) L (Ω)  3p   p    v N L22 (Ω) ∇ v N  L 22(Ω)   p   3p  C  v N  L22 (Ω) ∇ v N  L 2p (Ω) . 3p 

Thus, it suffices to have 2  p that is p  52 . Taking the weak limit of (4.16), we obtain

1 2

          ∂t | v |2 − div f D v N v + f D v N : D v N + v .∇ v . v + div( p v )   = div(τ v ) − τ N : D v N .

(4.17)

On the other hand, we take the weak limit of the first equation in ( S ) knowing that v N .∇ v N converges to v .∇ v

    + ∇ p = div τ , ∂t v + v .∇ v − div f D v N which is multiplied by v leads to

1 2

        ∂t | v |2 − div f D v N v + f D v N : D ( v ) + v .∇ v . v + div( p v ) = div(τ v ) − τ : D ( v ).

(4.18)

Subtracting (4.18) from (4.17), we get

 

f D vN



       : D v N − f D v N : D ( v ) = −τ N : D v N + τ : D ( v ).

Or equivalently

  

f D vN



           − f D (v ) : D (v ) − D v N = τ N − τ : D v N − D (v ) .

For the system ( S 2 ), under the hypotheses on f in Theorem 3.6, since f is monotone, we deduce that



  





τ N − τ : D v N − D ( v )  0,

and thus, the equation satisfied by ζ is reduced to

(∂t + v .∇)ζ 

2

(1 + η)2



  





τ N − τ : D v N − D ( v )  0, ζ (0, .) = 0,

which implies that ζ = η = 0. Now, for the system ( S 1 ), under the strong monotonicity condition on f in Theorem 3.5, we deduce that

               D v N − D ( v )2   τ N − τ : D v N − D ( v )   √η D v N − D ( v )2 12 2 1 1    η +  D v N − D ( v ) . 2

2

O. Bejaoui, M. Majdoub / J. Differential Equations 254 (2013) 660–685

681

Thus, we get

     D v N − D ( v )2  η, and

(∂t + v .∇)ζ  C ζ, which implies thanks to Gronwall’s Lemma and to the fact that ζ (0, .) = 0 that ζ = η = 0. Notice that if the quadratic term τ N . w ( v N ) − w ( v N ).τ N is present, then the difficulty is how to estimate | w ( v N ) − w ( v )|2 in terms of η . 4.6. Weak limits of non-linear terms involving D ( v N ) Under the hypotheses on f in Theorem 3.5, we have

     D v N − D ( v )2  η = 0. p

As | D ( v N ) − D ( v )|2 is bounded in L 2 ([0, T ] × Ω) with p > 2, then it is equi-integrable in L 1 ([0, T ] × Ω), and hence









D v N → D ( v ) strongly in L 2 [0, T ] × Ω . Vitali’s Lemma implies that

 

f D vN



  = f D (v ) ,

 

g D vN



  = g D (v ) .

Let us now focus on the system ( S 2 ). Under the hypotheses on f in Theorem 3.6, we have two ways to get f ( D ( v N )). Let us begin by the simplest way. Let μt ,x be the Young measure generated by the sequence D ( v N ) and let G (λ, t , x) := ( f (λ) − f ( D ( v )(t , x))) : (λ − D ( v )(t , x))  0. Remark that



 G (λ, t , x)μt ,x (λ) =

M n×n

lim inf δ→0

M n×n

G (λ, t , x) 1 + δ G (λ, t , x)

dμt ,x (λ).

By Fatou’s Lemma, we get



 G (λ, t , x) dμt ,x (λ)  lim inf δ→0

M n×n

M n×n

G (λ, t , x) 1 + δ G (λ, t , x)

dμt ,x (λ).

Notice that

G ( D ( v N ), t , x) 1 + δ G ( D ( v N ), t , x)



1

G ( D ( v N ), t , x)

δ 1+

G ( D ( v N ), t , x)

1

 , δ

δ ∈ ]0, 1].

G ( D ( v N ),t ,x) Hence, for a fixed δ ∈ ]0, 1], the sequence 1+δ G ( D ( v N ),t ,x) is bounded in L ∞ ([0, T ] × Ω) and thus equiintegrable in L 1 ([0, T ] × Ω).

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O. Bejaoui, M. Majdoub / J. Differential Equations 254 (2013) 660–685

We can claim that



G ( D ( v N ), t , x)

G (λ, t , x)

=

1 + δ G ( D ( v N ), t , x)

1 + δ G (λ, t , x)

M n×n

∀δ ∈ ]0, 1].

dμt ,x (λ),

Consequently, we get



G ( D ( v N ), t , x)

G (λ, t , x) dμt ,x (λ)  lim inf

1 + δ G ( D ( v N ), t , x)

δ→0

M n×n

     G D v N , t, x .

But, thanks to the strong convergence of τ N to τ in L 2 ([0, T ] × Ω) and to the weak convergence of D ( v N ) to D ( v ) in L 2 ([0, T ] × Ω), we deduce that



 







G (λ, t , x) dμt ,x (λ)  G D v N , t , x = −

 





τ N − τ D v N − D ( v ) = 0.

M n×n

This means that the div–curl inequality used in [8] and [9], which is the key ingredient to handle the limits of non-linear terms, remains true in our case and enables us to get f ( D ( v N )) = f ( D ( v )). The second way is to proceed as in [8] and [9] to get the div–curl inequality. As f is monotone, then the negative part of ( f ( D ( v N )) − f ( D ( v ))) : ( D ( v N ) − D ( v )) is null and thus weakly relatively compact in L 1 . Applying Theorem 3.4, we infer that

t 







   : λ − D ( v ) dμs,x (λ) dx ds

f (λ) − f D ( v )

0 Ω M n×n

t 

  

f D vN

 lim inf n→+∞



      − f D ( v ) : D v N − D ( v ) dx ds

0 Ω

t  n→+∞

  

f D vN

 lim sup



      − f D ( v ) : D v N − D ( v ) dx ds

0 Ω

t 







t 



f D ( v ) : D v N dx ds +

 − lim inf n→+∞

0 Ω

t



+ lim sup n→+∞





f D ( v ) : D ( v ) dx ds 0 Ω

 

f D v

N





:D v

N



t  n→+∞

0 Ω

 

f D vN

dx ds − lim inf



: D ( v ) dx ds.

0 Ω



The weak convergence in L p of D ( v N ) to D ( v ) and the fact that f ( D ( v )) belongs to L p imply that

t 









t 

f D ( v ) : D v N dx ds =

lim inf n→+∞

0 Ω





f D ( v ) : D ( v ) dx ds. 0 Ω

O. Bejaoui, M. Majdoub / J. Differential Equations 254 (2013) 660–685

683

Moreover, we have

t  lim inf

 

f D v

n→+∞

N



t 

 

: D ( v ) dx ds =

0 Ω

f D vN



: D ( v ) dx ds.

0 Ω

Testing the first equation in ( S ) by v N and integrating over [0, t ] × Ω , we get

t 

 

f D v

N





:D v

N



t   dx ds =

  N 2  N N   − ∂t v −τ : D v dx ds 1 2

0 Ω

0 Ω

t  −













1 1 2 2 τ N : D v N dx ds +  v 0N L 2 (Ω) −  v N (t , .) L 2 (Ω) . 2

2

0 Ω

The same arguments as for the classical Navier–Stokes equation lead to

v N (t , .)  v (t , .)

weakly in L 2 , ∀t ∈ [0, T ].

Thanks to the lower semi-continuity of the norm, we have

   v (t , .)

L 2 (Ω)

   lim inf v N (t , .) L 2 (Ω) . n→+∞

Using the fact that v 0N converges strongly to v 0 in L 2 , we obtain

t  n→+∞



 

f D vN

lim sup



  : D v N dx ds

0 Ω

t 

1

v 0 2L 2 (Ω) 2

+ lim sup n→+∞

2   1 −τ N : D v N dx ds −  v (t , .) L 2 (Ω) . 2

0 Ω

Knowing that

1 2

        ∂t | v |2 − div f D v N v + f D v N : D ( v ) + v .∇ v . v + div( p v ) = div(τ v ) − τ : D ( v ),

we get by integration over [0, t ] × Ω

t 

 

f D 0 Ω

Finally, we get

vN



1

2 1 : D ( v ) dx ds = −  v (t , .) L 2 (Ω) + v 0 2L 2 (Ω) − 2

t 

τ : D ( v ) dx ds.

2

0 Ω

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O. Bejaoui, M. Majdoub / J. Differential Equations 254 (2013) 660–685

t 







   : λ − D ( v ) dμs,x (λ) dx ds

f (λ) − f D ( v )

0 Ω M n×n

t 

  

f D vN

 lim inf n→+∞



      − f D ( v ) : D v N − D ( v ) dx ds

0 Ω

t   lim sup n→+∞



N

−τ : D v

N



t  dx ds +

0 Ω

τ : D ( v ) dx ds. 0 Ω

Since τ N converges strongly to τ in L 2 and D ( v N ) converges weakly in L 2 to D ( v ), then converges weakly in L 1 to τ : D ( v ), particularly

t 



N

τ :D v

lim

n→+∞

N



t  N

dx ds = lim sup n→+∞

0 Ω



τ :D v 0 Ω

N



τ N : D (v N )

t  dx ds =

τ : D ( v ) dx ds. 0 Ω

Hence

t 







   : λ − D ( v ) dμs,x (λ) dx ds

f (λ) − f D ( v )

0 Ω M n×n

t 

  

f D vN

 lim inf n→+∞



      − f D ( v ) : D v N − D ( v ) dx ds

0 Ω

 0. This ends the proofs of Theorems 3.5–3.6. Acknowledgment We are very grateful to Professor Nader Masmoudi for interesting discussions around the questions dealt with in this paper. References [1] H.W. Alt, Lineare Funktionalanalysis, second ed., Springer-Verlag, Berlin/Heidelberg/New York, 1992. [2] C. Andrade, Viscosity of liquids, Nature 125 (1930) 309–310. [3] N. Arada, A. Sequeira, Strong steady solutions for a generalized Oldroyd-B model with shear-dependent viscosity in a bounded domain, Math. Models Methods Appl. Sci. 13 (9) (2003) 1–21. [4] H. Brezis, Analyse fonctionnelle Théorie et applications, Masson, Paris, 1983. [5] P.W. Bridgman, The Physics of High Pressure, MacMillan, New York, 1931. [6] M. Bulicek, M. Majdoub, J. Malek, Unsteady flows of fluids with pressure dependent viscosity in unbounded domains, Nonlinear Anal. 11 (5) (2010) 3968–3983. [7] J.Y. Chemin, N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal. 33 (1) (2001) 84–112 (electronic). [8] P. Dreyfuss, N. Hungerbuhler, Results on a Navier–Stokes system with applications to electrorheological fluid flow, Int. J. Pure Appl. Math. 14 (2) (2004) 241–271. [9] P. Dreyfuss, N. Hungerbuhler, Navier–Stokes systems with quasimonotone viscosity tensor, Int. J. Differ. Equ. Appl. 9 (1) (2004) 59–79. [10] R.J. DiPerna, P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (3) (1989) 511–547.

O. Bejaoui, M. Majdoub / J. Differential Equations 254 (2013) 660–685

685

[11] E. Fernandez-Cara, F. Guillen, R.R. Ortega, Some theoretical results for viscoplastic and dilatant fluids with variable density, Nonlinear Anal. 28 (6) (1997) 1079–1100. [12] E. Fernandez-Cara, F. Guillen, R.R. Ortega, Some theoretical results concerning non-Newtonian fluids of the Oldroyd kind, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 26 (1) (1998) 1–29. [13] E. Fernandez-Cara, F. Guillen, R.R. Ortega, The mathematical analysis of viscoelastic fluids of the Oldroyd kind, 2000. [14] C. Guillopé, J.C. Saut, Existence results for the flow of viscoelastic fluids with a differential constitutive law, Nonlinear Anal. 15 (9) (1990) 849–869. [15] C. Guillopé, J.C. Saut, Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type, RAIRO Model. Math. Anal. Numer. 24 (3) (1990) 369–401. [16] R.R. Huilgol, Continuum Mechanics of Viscoelastic Liquids, Hindusthan Publishing Corporation, Delhi, 1975. [17] N. Hungerbuhler, A refinement of Ball’s Theorem on Young measures, New York J. Math. 3 (1997) 48–53. [18] R. Kupferman, C. Mangoubi, E.S. Titi, A Beale–Kato–Majda breakdown criterion for an Oldroyd-B fluid in the creeping flow regime, Math. Sci. 6 (1) (2008) 235–256. [19] Z. Lei, Rotation-strain decomposition for the incompressible viscoelasticity in two dimensions, http://arxiv.org/abs/ 1204.5763. [20] Z. Lei, Global existence of classical solutions for some Oldroyd-B model via the incompressible limit, Chinese Ann. Math. Ser. B 27 (5) (2006) 565–580. [21] Z. Lei, On 2D viscoelasticity with small strain, Arch. Ration. Mech. Anal. 198 (1) (2010) 13–37. [22] Z. Lei, C. Liu, Y. Zhou, Global existence for a 2D incompressible viscoelastic model with small strain, Commun. Math. Sci. 5 (3) (2007) 595–616. [23] Z. Lei, C. Liu, Y. Zhou, Global solutions for incompressible viscoelastic fluids, Arch. Ration. Mech. Anal. 188 (3) (2008) 371–398. [24] Z. Lei, N. Masmoudi, Y. Zhou, Remarks on the blowup criteria for Oldroyd models, J. Differential Equations 248 (2) (2010) 328–341. [25] Z. Lei, Y. Wang, Global solutions for micro–macro models of polymeric fluids, J. Differential Equations 250 (10) (2011) 3813–3830. [26] Z. Lei, Y. Zhou, Global existence of classical solutions for the two-dimensional Oldroyd model via the incompressible limit, SIAM J. Math. Anal. 37 (3) (2005) 797–814 (electronic). [27] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969. [28] P.L. Lions, N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chinese Ann. Math. Ser. B 21 (2) (2000) 131–146. [29] J. Malek, J. Necas, K.R. Rajagopal, Global analysis of the flows of fluids with pressure-dependent viscosities, Arch. Ration. Mech. Anal. 165 (3) (2002) 243–269. [30] J. Malek, J. Necas, M. Rokyta, M. Ruzicka, Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman–Hall, 1996. [31] J. Malek, K.R. Rajagopal, Mathematical properties of the solutions to the equations governing the flow of fluid with pressure and shear rate dependent viscosities, in: Handbook of Mathematical Fluid Dynamics, in: Handb. Differ. Equ., vol. 4, Elsevier/North-Holland, Amsterdam, 2007, pp. 407–444. [32] F. Murat, J.M. Ball, Remarks on Chacon’s biting lemma, Proc. Amer. Math. Soc. 107 (3) (1989) 655–663. ˇ [33] T. Rubicek, A generalization of the Lions–Temam compact imbedding theorem, Cas. Pˇest. Mat. 115 (1990) 338–342. [34] W.R. Schowalter, Mechanics of Non-Newtonian Fluids, Pergamon Press, Oxford, 1978. [35] A.Z. Szeri, Fluid Film Lubrication: Theory and Design, Cambridge University Press, 1998. [36] G.B. Thurston, Viscoelasticity of human blood, Biophys. J. 12 (1972) 1205–1217. [37] W. Walter, Differential and Integral Inequalities, Springer-Verlag, Berlin/Heidelberg/New York, 1970. [38] K. Yosida, Functional Analysis, Springer-Verlag, Berlin/Gottingen/Heidelberg, 1965. [39] E. Zeidler, Nonlinear Functional Analysis II/A-Linear Monotone Operators, Springer-Verlag, Berlin/Heidelberg/New York, 1990.