# Study of Viscoplastic Flows Governed by the Norton-Hoff Operator

Authors: J. Ferchichi; I. Gaied
Journal: ISRN Applied Mathematics
 (2011)

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## Abstract

We deal with viscoplastic flows. The fluid motion is governed by the nonlinear
incompressible Norton-Hoff operator with homogeneous boundary conditions. We
provide the well-posedness of the model in the static case. The idea is based on the
crucial properties of the so-called compliance functional coupled with the min-max
theory relayed to a variational analysis of the associated Lagrangian.


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## Body

## 1. Introduction and Motivation

In this work, we deal with viscoplastic flows. The fluid’s motion is governed by the nonlinear incompressible static Norton-Hoff model with homogeneous Dirichlet boundary conditions. The main purpose of this paper is to establish the existence and uniqueness of the flows solution to the system𝒫. The result is argued in Theorem 3.7.The main difficulties in this analysis are(1)
the nonlinearity of the operator is extremely high(2)
the flow’s velocity is not enough regular which does not allow to adopt the tools of the classical analysis.Many authors were interested in the Norton-Hoff law. It was introduced by Norton [1] in order to describe the unidimensional creep of steel at high temperature and extended by Hoff [2] to the multidimensional solicitations. Daignieres et al. [3] has generalized the Norton-Hoff law in plasticity and viscoplasticity. Temam [4] has proved that the Prandtl-Reuss law of plasticity is derived from the Norton-Hoff law when the exponent of the material tends to one, and recently Ferchichi and Zolésio [5] has provided an identification model for a free-boundary problem of a non-Newtonian fluid coupled with the heat equation.The paper is organized as follows: in Section2, we introduce the static Norton-Hoff model. We provide the corresponding framework. We supply the equivalence between the norm induced by the set of admissible velocities and the usual one. In Section 3, we analyse the properties of the Norton-Hoff operator and we provide the well-posedness of the static case by using the min-max theory followed by the Lagrangian functional.


## 2. The Norton-Hoff Static Problem

### 2.1. Steady Incompressible Model

Consider a bounded open-domainΩ in ℝN,N=2,3, locally on one side of its C2-boundary ∂Ω, occupied by a viscoplastic fluid. The fluid motion is governed by the incompressible Norton-Hoff model with homogeneous boundary conditions to the velocity. The Norton-Hoff static problem consists in looking for a velocity field u defined on Ω¯ and fulfills the hereafter equations:(2.1)P{Kc|ɛ(u)|p-2ɛ(u)+PId=σ,inΩ,-div⁡(σ)=f,inΩ,div⁡(u)=0,inΩ,u=0,on⁡∂Ω,
where Kc is the consistency of the material, σ is the Cauchy stress tensor, ɛ(u)=(1/2)(D(u)+D(u)*) is the linearized strain velocity tensor, D is the differential operator, p is the exponent of the material; 1<p<2; it is the sensibility coefficient of the material to the strain velocity tensor, P is the hydrostatic pressure, Id is the identity tensor, and f is the density of the gravitation acting on the fluid. The first equation designates the behavior law, the second one describes the equilibrium state, and the third one prescribes the incompressibility of the fluid during the evolution.


### 2.2. Functional Setting

Let us introduce the functional framework(2.2)W=W01,p(Ω,RN)3,Wdiv⁡={v∈W,s.t.div⁡(v)=0on⁡∂Ω},L02(Ω)=L2(Ω)R.Further, we introduce the set of the admissible right-hand side(2.3)W-1,p′(Ω,RN)3,which is the topological dual space of𝒲 endowed with its natural norm, where p′ is the algebric dual of p. We define 〈·,·〉=〈·,·〉V*,V with V* being the topological dual space of V.The following result provides a Banach space structure for the spaces𝒲 and 𝒲div.Proposition 2.1.
The mapping∥·∥ defined from 𝒲 to ℝ+ by
(2.4)‖v‖=(∫Ω|ɛ(v)|p)1/pis a norm on 𝒲, where |·| means the Euclidian norm of a matrix in ℝN2.Poincare’s and Korn’s inequalities prove the equivalence between this norm and the usual one (see [6]). All the other functional spaces are used with their natural structure, unless mentioned.Proposition 2.2 (Poincare’s Inequality).
For any adequate domainΩ, there exists a constant cp(Ω) such that, for any v in 𝒲,
(2.5)‖v‖Lp(Ω,RN)≤cp(Ω)‖Dv‖Lp(Ω,R2N).Proof.
Since this proposition is classical, we briefly give the main steps of the proof. We consider, for any nonzerov in 𝒲, the quotient
(2.6)Q(v)=‖Dv‖Lpp‖v‖Lpp.
This functional has a minimum over𝒲/{0} which coincides with the minimum over the subset B of elements of 𝒲 with Lp-norm equals to 1. Considering a minimizing sequence (vn)n∈ℕ in B, it is bounded in 𝒲 so it converges weakly towards a v*. Due to the lower semicontinuity in Lp-norm, if Q(v*)≠0, then cp(Ω)=Q(v*)-1 is convenient. This holds since Q(v*)=0 if and only if v* is a non-zero constant, which is not possible because v=0 on the boundary.Proposition 2.3 (Korn’s Inequality).
For any adequate domainΩ, there exists a constant cK(Ω), such that, for any v in 𝒲,
(2.7)‖Dv‖Lp(Ω,RN2)≤cK(Ω)(‖ɛ(v)‖Lp(Ω,RN2)+‖v‖Lp(Ω,RN)).We refer to the study of functional spaces for Norton-Hoff materials made by Geymonat and Suquet in [6] and for the proof of Korn’s inequality. It is important to notice that this inequality does not hold for p=1 (see a counterexample of Orstein in [7]). The proof of Poincare’s inequality is classical and may be found in [4].We can now provide the proof of Proposition2.1.Proof of Proposition2.1.
For shortness, we denote∥·∥1,p=∥·∥W1,p(Ω,ℝN) the usual norm of W1,p(Ω,ℝN).
For anyv in 𝒲, we have
(2.8)‖v‖=12‖Dv+*Dv‖Lp(Ω,RN2)≤‖Dv‖Lp(Ω,RN2),‖v‖≤‖Dv‖Lp(Ω,RN2)+‖v‖Lp(Ω,RN)=‖v‖1,p,which proves that W1,p(Ω,ℝN)⊂𝒲(Ω). The converse inclusion is a mere consequence of Korn’s inequality.
We assume that there exists a sequence(vn)n∈ℕ in 𝒲 and a sequence (cn)n∈ℕ in ℝ+ which tends to infinity such that ∥vn∥1,p≥cn∥vn∥. Without loss of generality, we suppose ∥vn∥Lp(Ω,ℝN)=1. By Korn’s inequality, we have cK(Ω)∥vn∥+cK(Ω)+1≥cn∥vn∥ for any n. Hence,
(2.9)cK(Ω)+1cn-cK(Ω)≥‖vn‖,cK(Ω)(cK(Ω)+1)cn-cK(Ω)+cK(Ω)≥‖Dvn‖Lp(Ω,RN2).
Thus, the sequence(vn)n∈ℕ is bounded in 𝒲 endowed with its classical topology. So it, up a subsequence, converges weakly towards v* in 𝒲. Since ∥·∥ is lower semicontinuous,
(2.10)‖v*‖≤lim⁡⁡inf⁡n→∞⁡‖vn‖=0.
So,v*=0 which is contradictory with ∥vn∥Lp(Ω,ℝN)=1. Hence, there exists a constant c such that
(2.11)∀v∈W,‖v‖1,p≤c‖v‖.
Thus, the equivalence is provided.


## 2.1. Steady Incompressible Model

Consider a bounded open-domainΩ in ℝN,N=2,3, locally on one side of its C2-boundary ∂Ω, occupied by a viscoplastic fluid. The fluid motion is governed by the incompressible Norton-Hoff model with homogeneous boundary conditions to the velocity. The Norton-Hoff static problem consists in looking for a velocity field u defined on Ω¯ and fulfills the hereafter equations:(2.1)P{Kc|ɛ(u)|p-2ɛ(u)+PId=σ,inΩ,-div⁡(σ)=f,inΩ,div⁡(u)=0,inΩ,u=0,on⁡∂Ω,
where Kc is the consistency of the material, σ is the Cauchy stress tensor, ɛ(u)=(1/2)(D(u)+D(u)*) is the linearized strain velocity tensor, D is the differential operator, p is the exponent of the material; 1<p<2; it is the sensibility coefficient of the material to the strain velocity tensor, P is the hydrostatic pressure, Id is the identity tensor, and f is the density of the gravitation acting on the fluid. The first equation designates the behavior law, the second one describes the equilibrium state, and the third one prescribes the incompressibility of the fluid during the evolution.


## 2.2. Functional Setting

Let us introduce the functional framework(2.2)W=W01,p(Ω,RN)3,Wdiv⁡={v∈W,s.t.div⁡(v)=0on⁡∂Ω},L02(Ω)=L2(Ω)R.Further, we introduce the set of the admissible right-hand side(2.3)W-1,p′(Ω,RN)3,which is the topological dual space of𝒲 endowed with its natural norm, where p′ is the algebric dual of p. We define 〈·,·〉=〈·,·〉V*,V with V* being the topological dual space of V.The following result provides a Banach space structure for the spaces𝒲 and 𝒲div.Proposition 2.1.
The mapping∥·∥ defined from 𝒲 to ℝ+ by
(2.4)‖v‖=(∫Ω|ɛ(v)|p)1/pis a norm on 𝒲, where |·| means the Euclidian norm of a matrix in ℝN2.Poincare’s and Korn’s inequalities prove the equivalence between this norm and the usual one (see [6]). All the other functional spaces are used with their natural structure, unless mentioned.Proposition 2.2 (Poincare’s Inequality).
For any adequate domainΩ, there exists a constant cp(Ω) such that, for any v in 𝒲,
(2.5)‖v‖Lp(Ω,RN)≤cp(Ω)‖Dv‖Lp(Ω,R2N).Proof.
Since this proposition is classical, we briefly give the main steps of the proof. We consider, for any nonzerov in 𝒲, the quotient
(2.6)Q(v)=‖Dv‖Lpp‖v‖Lpp.
This functional has a minimum over𝒲/{0} which coincides with the minimum over the subset B of elements of 𝒲 with Lp-norm equals to 1. Considering a minimizing sequence (vn)n∈ℕ in B, it is bounded in 𝒲 so it converges weakly towards a v*. Due to the lower semicontinuity in Lp-norm, if Q(v*)≠0, then cp(Ω)=Q(v*)-1 is convenient. This holds since Q(v*)=0 if and only if v* is a non-zero constant, which is not possible because v=0 on the boundary.Proposition 2.3 (Korn’s Inequality).
For any adequate domainΩ, there exists a constant cK(Ω), such that, for any v in 𝒲,
(2.7)‖Dv‖Lp(Ω,RN2)≤cK(Ω)(‖ɛ(v)‖Lp(Ω,RN2)+‖v‖Lp(Ω,RN)).We refer to the study of functional spaces for Norton-Hoff materials made by Geymonat and Suquet in [6] and for the proof of Korn’s inequality. It is important to notice that this inequality does not hold for p=1 (see a counterexample of Orstein in [7]). The proof of Poincare’s inequality is classical and may be found in [4].We can now provide the proof of Proposition2.1.Proof of Proposition2.1.
For shortness, we denote∥·∥1,p=∥·∥W1,p(Ω,ℝN) the usual norm of W1,p(Ω,ℝN).
For anyv in 𝒲, we have
(2.8)‖v‖=12‖Dv+*Dv‖Lp(Ω,RN2)≤‖Dv‖Lp(Ω,RN2),‖v‖≤‖Dv‖Lp(Ω,RN2)+‖v‖Lp(Ω,RN)=‖v‖1,p,which proves that W1,p(Ω,ℝN)⊂𝒲(Ω). The converse inclusion is a mere consequence of Korn’s inequality.
We assume that there exists a sequence(vn)n∈ℕ in 𝒲 and a sequence (cn)n∈ℕ in ℝ+ which tends to infinity such that ∥vn∥1,p≥cn∥vn∥. Without loss of generality, we suppose ∥vn∥Lp(Ω,ℝN)=1. By Korn’s inequality, we have cK(Ω)∥vn∥+cK(Ω)+1≥cn∥vn∥ for any n. Hence,
(2.9)cK(Ω)+1cn-cK(Ω)≥‖vn‖,cK(Ω)(cK(Ω)+1)cn-cK(Ω)+cK(Ω)≥‖Dvn‖Lp(Ω,RN2).
Thus, the sequence(vn)n∈ℕ is bounded in 𝒲 endowed with its classical topology. So it, up a subsequence, converges weakly towards v* in 𝒲. Since ∥·∥ is lower semicontinuous,
(2.10)‖v*‖≤lim⁡⁡inf⁡n→∞⁡‖vn‖=0.
So,v*=0 which is contradictory with ∥vn∥Lp(Ω,ℝN)=1. Hence, there exists a constant c such that
(2.11)∀v∈W,‖v‖1,p≤c‖v‖.
Thus, the equivalence is provided.


## 3. Properties of the Norton-Hoff Operator

Definition 3.1.
The Norton-Hoff functional or the so-called compliance functional is given by(3.1)ΦΩ:W(Ω)⟶R,v⟶∫ΩKcp|ɛ(v)|p-fv.The existence and uniqueness of a solution to the Norton-Hoff problem𝒫 is derived from the properties of ΦΩ. As a preliminary result, we have the following differentiability property.Lemma 3.2.
The functionalJ:v→∥v∥p is Gateaux differentiable in 𝒲. Its Gateaux derivative at a point u in direction v is given by
(3.2)J′(u;v)=∫Ωp|ɛ(u)|p-2ɛ(u)⋯ɛ(v),where the expression |ɛ(u)|p-2ɛ(u)⋯ɛ(v) is to be understood as continuously extended with 0 at any point x with |ɛ(u)|(x)=0.Proof.
Letu and v be in 𝒲. Let λ be a positive real number. The differential quotient is
(3.3)J(u+λv)-J(u)λ=1λ∫Ω(|ɛ(u)+λɛ(v)|p-|ɛ(u)|p).
The functionρ:X→Xp is differentiable on ℝ+, with derivative ρ′(X)=pXp-1 for X>0 and ρ′(0)=0. Hence, for almost every point x in Ω, we can apply Taylor’s formula in order to get
(3.4)|ρ(|ɛ(u)(x)+λɛ(v)(x)|)-ρ(|ɛ(u)(x)|)λ|≤p(|ɛ(u)(x)|+λ|ɛ(v)(x)|)p-1|ɛ(v)(x)|.
If we assume thatλ∈]0,1], then we can overestimate this quotient,
(3.5)|ρ(|ɛ(u)(x)+λɛ(v)(x)|)-ρ(|ɛ(u)(x)|)λ|≤p(|ɛ(u)(x)|+|ɛ(v)(x)|)p-1|ɛ(v)(x)|.
So, Lebesgue’s theorem supplies [8]:
(3.6)lim⁡λ→0⁡J(u+λv)-J(u)λ=∫Ωlim⁡λ→0⁡1λ(|ɛ(u)+λɛ(v)|p-|ɛ(u)|p).
Because of(3.7)|ɛ(u)(x)+λɛ(v)(x)|2-|ɛ(u)(x)|2λ=λɛ(v)⋯ɛ(v)+ɛ(u)⋯ɛ(v),the functionu→|ɛ(u)|2 is Gateaux differentiable and its Gateaux derivative at u in a direction v is 2ɛ(u)⋯ɛ(v). By composition with the real function Y→Yp/2, we come to
(3.8)lim⁡λ→0⁡|ɛ(u)+λɛ(v)|p-|ɛ(u)(x)|pλ=p|ɛ(u)|p-2ɛ(u)⋯ɛ(v),which achieves the proof.Remark 3.3.
Notice that we also can prove the Fréchet differentiability of the norm, with derivative that coincides (up to the coefficientKc/p) with the Norton-Hoff operator.Proposition 3.4.
The functionalΦΩ is strictly convex, weakly l.s.c., Gateaux differentiable, and coercive. Moreover, for any u and v in 𝒲(Ω), its directional derivative at u in direction v is given by
(3.9)Φ′Ω(u,v)=∫ΩKc|ɛ(u)|p-2ɛ(u)⋯ɛ(v)-fv.Proof.
SinceΦΩ(v)=(Kc/p)∥v∥p-〈f,v〉𝒲′×𝒲, the continuity for the strong topology is obvious  and so is the lower semi-continuity (l.s.c.) for the weak topology of 𝒲. By this equality, the strict convexity of ΦΩ derives from the strict convexity of ∥·∥p, which is a mere consequence of both ∥·∥ and x→xp from ℝ+ to itself are convex functions.
The coercivity ofΦΩ comes from
(3.10)∀v∈W,ΦΩ(v)≥Kcp‖v‖p-‖f‖W'‖v‖W.
Letu and v be in 𝒲 and let λ be a positive number. For any x∈Ω, we denote
(3.11)Qλ(x)=|ɛ(u)(x)+λɛ(v)(x)|p-|ɛ(u)(x)|pλ.
Using the convexity ofX→|X| from the space of n×n matrices to ℝ and the convexity of x→xp, we have
(3.12)Qλ(x)=|λ(ɛ(u)(x)+ɛ(v)(x))+(1-λ)ɛ(u)(x)|p-|ɛ(u)(x)|pλ≤λ|ɛ(u+v)(x)|p+(1-λ)|ɛ(u)(x)|p-|ɛ(u)(x)|pλ≤|ɛ(u+v)(x)|p-|ɛ(u)(x)|p=Q1(x).
Moreover, classical calculus yields(3.13)lim⁡λ→0⁡Qλ(x)=p|ɛ(u)(x)|p-2ɛ(u)(x)⋯ɛ(v)(x),which is continuously extended by 0 if ɛ(u)(x)=0.
Since all theQλ belong to L1(Ω), we have
(3.14)lim⁡λ→0⁡ΦΩ(u+λv)-ΦΩ(u)λ=lim⁡λ→0⁡∫ΩKcpQλ(x)dx-〈f,v〉W'×W=∫Ωlim⁡λ→0⁡KcpQλ(x)dx-〈f,v〉W'×W=∫ΩKc|ɛ(u)(x)|p-2ɛ(u)(x)⋯ɛ(v)(x)-fv.
Hence, we get the proof.


### 3.1. Well-Posedness via Min-Max Theory

In order to look for a solution to the Norton Hoff problem𝒫, we adopt the min-max theory (see [9]); mainly we use the following theorem.Theorem 3.5.
LetE and  F be two reflexive spaces and  L a function from  E×F to  ℝ. One assumes the following: (i)
For anya in E, the function L(a,·) is concave and upper semi-continuous (u.s.c).(ii)
For anyb in F, the function L(·,b) is convex and lower semi-continuous (l.s.c.).(iii)
There existsb0 in F such that
(3.15)lim⁡∥a∥→∞⁡L(a,b0)=+∞.(iv)
(3.16)lim⁡∥b∥→∞⁡inf⁡a∈E⁡L(a,b)=-∞.
Then,L admits a saddle point (a¯,b¯), that is,
(3.17)∀(a,b)inE×F;L(a¯,b)≤L(a¯,b¯)≤L(a,b¯).
Moreover, the set of saddle points is convex and is a Cartesian product(i.e.,if(a¯1,b¯1)and(a¯2,b¯2)aresaddlepointsofL,thensodo(a¯1,b¯2)and(a¯2,b¯1)).We refer to [4] for the proof.LetL be the following Lagrangian associated with the Norton-Hoff operator:
(3.18)L:W×Q⟶R,(v,q)⟶∫Ω(Kcp|ɛ(v)|p-fx-qdiv⁡⁡v).Proposition 3.6.
The functionL has at least a saddle point.Proof.
It is enough to apply Theorem3.5 to the previous Lagrangian associated with our problem. Since the operator L is a linear with respect to q, condition (i) obviously holds.
Condition (ii) is satisfied. In fact, for anyq in Q, L(·,q) is the sum of ΦΩ and a linear functional; hence, with the fact that the mapping ΦΩ is convex and lower semi-continuous, we get the assertion.
The choiceq0=0 is convenient for condition (iii) since L(·,0) is coercive.
We end the proof by noticing that(3.19)lim⁡‖q‖↑∞⁡inf⁡v∈W⁡L(v,q)≤lim⁡‖q‖↑∞⁡(inf⁡v∈W⁡ΦΩ(v)+inf⁡v∈W⁡-∫Ωqdiv⁡⁡v)≤inf⁡v∈W⁡ΦΩ(v)+lim⁡∥q∥↑∞⁡(inf⁡v∈W⁡-〈q,div⁡⁡v〉).
Whileinfv∈𝒲ΦΩ(v)≤ΦΩ(0)=0, then
(3.20)inf⁡v∈W⁡ΦΩ(v)+lim⁡∥q∥↑∞⁡inf⁡v∈W⁡-〈q,div⁡⁡v〉=-∞,which achieves the proof of the existence for a saddle point.Theorem 3.7.
The Norton-Hoff mixte problem𝒫 has a unique solution  (u,P) that belongs to  𝒲×L02.Proof.
For the existence, it is sufficient to prove that a saddle point of the operatorL provides a solution to the problem 𝒫.
Let(u,P) be a saddle point of L, then one has
(3.21)∀(v,q)∈W×L02;L(u,q)≤L(u,P)≤L(v,P).
The first inequality yields(3.22)∫ΩKcp|ɛ(u)|p-fu-qdiv⁡⁡u≤∫ΩKcp|ɛ(u)|p-fu-Pdiv⁡⁡v,which holds for any q in L02. Hence, divu=0 so u belongs to 𝒲div.
We notice that, as a simple consequence of the announced properties ofΦΩ the Lagrangian L is Gateaux differentiable both with respect to its first and second variable. According to [9], first-order optimality conditions hold. As a consequence, the second inequality implies that, for any v in 𝒲, the Gateaux derivative of L at point (u,P) in direction v with respect to the first variable vanishes.
Hence,(3.23)∫ΩKcp|ɛ(u)|p-2ɛ(u)⋯ɛ(v)-fv-Pdiv⁡⁡v=0.
Using Green’s formula, see [8], we come to
(3.24)〈-div⁡(Kcp|ɛ(u)|p-2ɛ(u))-fu+∇P,v〉W′×W=0,which is the weak formulation of the problem 𝒫 for any v in 𝒲. Eventually, we have a solution of 𝒫 in 𝒲×L02.
The converse is easily proven as a mere application of Green’s formula: any solution of𝒫 in 𝒲×Q is a saddle point of L.
Consequently, in order to provide the uniqueness of solution, we proceed as follows. If(u′,P′) is another solution of 𝒫, then it is a saddle point of L. It is well known that the set of saddle points is a cartesian product (see, e.g., [9]). Hence, (u′,P) is a third saddle point of L. But both u and u′ are solution of
(3.25)min⁡v∈Wdiv⁡⁡ΦΩ(v).
OrΦΩ is strictly convex, continuous, and coercive functional, so it has a unique minimum. One can deduce that u′ = u. If we assume that (u,P) and (u′,P′) are both solutions of 𝒫, then we come to ∇(P-P′)=0 in Ω. Whence P and P′ are equal; up to a constant that is, they are equal in L02.
This achieves the proof.


## 3.1. Well-Posedness via Min-Max Theory

In order to look for a solution to the Norton Hoff problem𝒫, we adopt the min-max theory (see [9]); mainly we use the following theorem.Theorem 3.5.
LetE and  F be two reflexive spaces and  L a function from  E×F to  ℝ. One assumes the following: (i)
For anya in E, the function L(a,·) is concave and upper semi-continuous (u.s.c).(ii)
For anyb in F, the function L(·,b) is convex and lower semi-continuous (l.s.c.).(iii)
There existsb0 in F such that
(3.15)lim⁡∥a∥→∞⁡L(a,b0)=+∞.(iv)
(3.16)lim⁡∥b∥→∞⁡inf⁡a∈E⁡L(a,b)=-∞.
Then,L admits a saddle point (a¯,b¯), that is,
(3.17)∀(a,b)inE×F;L(a¯,b)≤L(a¯,b¯)≤L(a,b¯).
Moreover, the set of saddle points is convex and is a Cartesian product(i.e.,if(a¯1,b¯1)and(a¯2,b¯2)aresaddlepointsofL,thensodo(a¯1,b¯2)and(a¯2,b¯1)).We refer to [4] for the proof.LetL be the following Lagrangian associated with the Norton-Hoff operator:
(3.18)L:W×Q⟶R,(v,q)⟶∫Ω(Kcp|ɛ(v)|p-fx-qdiv⁡⁡v).Proposition 3.6.
The functionL has at least a saddle point.Proof.
It is enough to apply Theorem3.5 to the previous Lagrangian associated with our problem. Since the operator L is a linear with respect to q, condition (i) obviously holds.
Condition (ii) is satisfied. In fact, for anyq in Q, L(·,q) is the sum of ΦΩ and a linear functional; hence, with the fact that the mapping ΦΩ is convex and lower semi-continuous, we get the assertion.
The choiceq0=0 is convenient for condition (iii) since L(·,0) is coercive.
We end the proof by noticing that(3.19)lim⁡‖q‖↑∞⁡inf⁡v∈W⁡L(v,q)≤lim⁡‖q‖↑∞⁡(inf⁡v∈W⁡ΦΩ(v)+inf⁡v∈W⁡-∫Ωqdiv⁡⁡v)≤inf⁡v∈W⁡ΦΩ(v)+lim⁡∥q∥↑∞⁡(inf⁡v∈W⁡-〈q,div⁡⁡v〉).
Whileinfv∈𝒲ΦΩ(v)≤ΦΩ(0)=0, then
(3.20)inf⁡v∈W⁡ΦΩ(v)+lim⁡∥q∥↑∞⁡inf⁡v∈W⁡-〈q,div⁡⁡v〉=-∞,which achieves the proof of the existence for a saddle point.Theorem 3.7.
The Norton-Hoff mixte problem𝒫 has a unique solution  (u,P) that belongs to  𝒲×L02.Proof.
For the existence, it is sufficient to prove that a saddle point of the operatorL provides a solution to the problem 𝒫.
Let(u,P) be a saddle point of L, then one has
(3.21)∀(v,q)∈W×L02;L(u,q)≤L(u,P)≤L(v,P).
The first inequality yields(3.22)∫ΩKcp|ɛ(u)|p-fu-qdiv⁡⁡u≤∫ΩKcp|ɛ(u)|p-fu-Pdiv⁡⁡v,which holds for any q in L02. Hence, divu=0 so u belongs to 𝒲div.
We notice that, as a simple consequence of the announced properties ofΦΩ the Lagrangian L is Gateaux differentiable both with respect to its first and second variable. According to [9], first-order optimality conditions hold. As a consequence, the second inequality implies that, for any v in 𝒲, the Gateaux derivative of L at point (u,P) in direction v with respect to the first variable vanishes.
Hence,(3.23)∫ΩKcp|ɛ(u)|p-2ɛ(u)⋯ɛ(v)-fv-Pdiv⁡⁡v=0.
Using Green’s formula, see [8], we come to
(3.24)〈-div⁡(Kcp|ɛ(u)|p-2ɛ(u))-fu+∇P,v〉W′×W=0,which is the weak formulation of the problem 𝒫 for any v in 𝒲. Eventually, we have a solution of 𝒫 in 𝒲×L02.
The converse is easily proven as a mere application of Green’s formula: any solution of𝒫 in 𝒲×Q is a saddle point of L.
Consequently, in order to provide the uniqueness of solution, we proceed as follows. If(u′,P′) is another solution of 𝒫, then it is a saddle point of L. It is well known that the set of saddle points is a cartesian product (see, e.g., [9]). Hence, (u′,P) is a third saddle point of L. But both u and u′ are solution of
(3.25)min⁡v∈Wdiv⁡⁡ΦΩ(v).
OrΦΩ is strictly convex, continuous, and coercive functional, so it has a unique minimum. One can deduce that u′ = u. If we assume that (u,P) and (u′,P′) are both solutions of 𝒫, then we come to ∇(P-P′)=0 in Ω. Whence P and P′ are equal; up to a constant that is, they are equal in L02.
This achieves the proof.


## 4. Conclusion

We have studied the problem of well-posedness of the Norton-Hoff operator that models visco-plastic flows. We have supplied an existence and uniqueness result to the considered system. We were faced by the fact that the flow is governed by a less-regular velocity field. Nevertheless, we have came over to this difficulty by using the theory of min-max applied to the so-called compliance functional. The results that were obtained encourage further analysis of the proposed techniques including identification of free interfaces.



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*Source: 627318-2011-07-07.xml*